Answers (Anticipation Guide and Lesson 10-1)

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1 Answers (Anticipation Guide and Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Lesson Reading Guide Midpoint and Distance Formulas Get Read for the Lesson Read the introduction to Lesson 0- in our tetbook. How do ou find distances on a road map? Sample answer: Use the scale of miles on the map. You might also use a ruler. Read the Lesson. a. Write the coordinates of the midpoint of a segment with endpoints (, ) and (, )., b. Eplain how to find the midpoint of a segment if ou know the coordinates of the endpoints. Do not use subscripts in our eplanation. Sample answer: To find the -coordinate of the midpoint, add the -coordinates of the endpoints and divide b two. To find the -coordinate of the midpoint, do the same with the -coordinates of the endpoints.. a. Write an epression for the distance between two points with coordinates (, ) and (, ). ( ) ( ) b. Eplain how to find the distance between two points. Do not use subscripts in our eplanation. Sample answer: Find the difference between the -coordinates and square it. Find the difference between the -coordinates and square it. Add the squares. Then find the square root of the sum. 3. Consider the segment connecting the points (3, ) and (, ). a. Find the midpoint of this segment. (3, ) b. Find the length of the segment. Write our answer in simplified radical form. 6 Remember What You Learned. How can the mid in midpoint help ou remember the midpoint formula? Sample answer: The midpoint is the point in the middle of a segment. It is halfwa between the endpoints. The coordinates of the midpoint are found b finding the average of the two -coordinates (add them and divide b ) and the average of the two -coordinates. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID 0 Anticipation Guide Conic Sections Step Before ou begin Chapter 0 Read each statement. Decide whether ou Agree (A) or Disagree (D) with the statement. Write A or D in the first column R if ou are not sure whether ou agree or disagree, write NS (Not Sure). STEP STEP Statement A, D, or NS A or D After ou complete Chapter 0 Reread each statement and complete the last column b entering an A or a D. Did an of our opinions about the statements change from the first column? For those statements that ou mark with a D, use a piece of paper to write an eample of wh ou disagree. Chapter 0 3 Glencoe Algebra Chapter 0 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter Resources. To find the midpoint between two points on a coordinate plane, find the means of the coordinates of the points.. All points on a parabola are the same distance from a given point and the -ais. 3. In the equation for a parabola, a( h) k, if a 0 the parabola opens downward.. If the equation of a circle is ( 3) ( 6) 6, then the center of the circle is at the point (3, 6).. A tangent line to a circle is a line that intersects the circle in two points. 6. The foci of an ellipse alwas lie on the major ais. A 7. The asmptotes of a graph of a hperbola are lines that the graph approaches but never reaches.. Conic sections are formed b the intersection of two cones. D. When the equation of a conic section is written in standard form it is possible to name which conic section the equation represents. 0. No solution can be found for a sstem of equations containing both linear and quadratic equations. A D A D D A A D Step Chapter 0 A Glencoe Algebra

2 Answers (Lesson 0-) Lesson 0- NAME DATE PERID 0- Stud Guide and Intervention (continued) Midpoint and Distance Formulas The Distance Formula Distance Formula The distance between two points (, ) and (, ) is given b d ( ) (. ) Eample What is the distance between (, ) and (6, )? d ( ( ) ) Distance Formula (6 ) [ ()] Let (, ) (, ) and (, ) (6, ). () 6) ( Subtract or 3 Simplif. The distance between the points is 3 or about. units. Eample Find the perimeter and area of square PQRS with vertices P(, ), Q(, 7), R(, ), and S(, ). Find the length of one side to find the perimeter and the area. Choose PQ. d ( ( ) ) Distance Formula [ ()] ( 7) Let (, ) (, ) and (, ) (, 7). () ) (6 Subtract. 0 or 0 Simplif. Since one side of the square is 0, the perimeter is 0 units. The area is (0),or 0 units. Eercises Find the distance between each pair of points with the given coordinates.. (3, 7) and (, ). (, 0) and (0, ) 3. (6, 6) and (, 0) units 3 units 0 units. (7, ) and (, ). (, ) and (3, ) 6. (, ) and (6, ) 3 units 0 units units 7. (3, ) and (6, ). (3, ) and (, ). (, 7) and (, ) 33 units 0 units 6 units 0. Rectangle ABCD has vertices A(, ), B(3, ), C(3, ), and D(, ). Find the perimeter and area of ABCD. 3 6 units; 36 units. Circle R has diameter ST with endpoints S(, ) and T(, 3). What are the circumference and area of the circle? (Epress our answer in terms of.) 0 units; units Chapter 0 7 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID 0- Stud Guide and Intervention Midpoint and Distance Formulas The Midpoint Formula Midpoint Formula The midpoint M of a segment with endpoints (, ) and (, ) is,. Eample Eample Find the midpoint of the line segment with endpoints at (, 7) and (, 3).,, (), 7 3 or (, ) The midpoint of the segment is (, ). A diameter AB of a circle has endpoints A(, ) and B(7, 6). What are the coordinates of the center of the circle? The center of the circle is the midpoint of all of its diameters.,, (7), or, The circle has center,. 6 Eercises Find the midpoint of each line segment with endpoints at the given coordinates.. (, 7) and (, ). (, 3) and (0, ) 3. (, ) and (0, ) (, ) (, 3) (7, ). (3, 3) and (3, 3). (, 6) and (, ) 6. (, ) and (0, 6) (0, 0) (3., 0) (6, ) 7. (3, ) and (6, ) 3. (, ) and (7, 3). (., 6.) and (7., 3.7),, (., 3.) 0. (7, 6) and (, ). (3, 0) and (30, 0). (,.7) and (,.3) 33 (, ), (0,.) 3. Segment MN has midpoint P. If M has coordinates (, 3) and P has coordinates (, 6), what are the coordinates of N? (30, ). Circle R has a diameter ST. If R has coordinates (, ) and S has coordinates (, ), what are the coordinates of T? (, 0). Segment AD has midpoint B, and BD has midpoint C. If A has coordinates (, ) and C has coordinates (0, ), what are the coordinates of B and D? B is, 3, D is, 3 3. Chapter 0 6 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A Glencoe Algebra

3 Answers (Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Practice Midpoint and Distance Formulas Find the midpoint of each line segment with endpoints at the given coordinates.. (, 3), (6, ) (, 7). (, ), (0, 6), 3. (7, 6), (, ) (3, ). (, ), (, ) (, ). (, ), (, ), 7 6. (3, 3), (, ), (, ), (3, 7),. (6, 7), (, ),. (, ), (, ) (6, 0) 0. (, ), (3, 7),. (6, 3), (, 7),. (, ), (, 3), 3 3. (.6,.7), (.,.) (.,.).,6, 3, 6,. (.,.), (.,.) (., 0) 6.,,,,0 Find the distance between each pair of points with the given coordinates. 7. (, ), (, ) units. (, ), (, ) 0 units. (3, ), (, ) 73 units 0. (0, ), (, 6) 30 units. (, 6), (6, 6) unit. (3, ), (, 3) 7 units 3. (, 3), (, 3) 7 units. (, ), (7, ) 6 units. (, 3), (, ) units 6. (, 7), (0, 6) 70 units 7. (0, 3), (, ) 3 units. (0., 6), (., 0) 0 units 3 7.,,, unit 30. (, ), (, ) 7 units Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- Skills Practice Chapter 0 Glencoe Algebra 3. GEMETRY Circle has a diameter AB. If A is at (6, ) and B is at (3, ), find the center of the circle and the length of its diameter.,; 3 units 3. GEMETRY Find the perimeter of a triangle with vertices at (, 3), (, ), and (, ). 7 units Chapter 0 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Midpoint and Distance Formulas Find the midpoint of each line segment with endpoints at the given coordinates.. (, ), (, ) (0, 0). (, ), (, ) (, 3) 3. (3, ), (, ) (, ). (6, ), (, ), 3. (3, ), (, 3),3 6. (3, ), (3, ) 3, 7. (3, ), (, 0) (, ). (3, ), (, ) (, ). (, ), (, ) 0, 0. (, ), (0, ),. (3, 6), (, 3),. (0, 0), (, ), Find the distance between each pair of points with the given coordinates. 3. (, ), (, 0) 3 units. (7, 7), (, ) units. (, ), (, ) units 6. (, ), (, ) units 7. (, 6), (7, ) 0 units. (3, ), (3, ) units. (, 3), (3, ) units 0. (, 3), (, 7) units. (, ), (3, 0) 7 units. (3, ), (, 3) 3 units 3. (6, ), (, 3) 7 units. (, ), (, ) 6 units. (0, 3), (, ) units 6. (, 6), (, 0) units Chapter 0 A3 Glencoe Algebra

4 Answers (Lesson 0-) Lesson 0-0- NAME DATE PERID Enrichment Distance Between Points in Space The Distance Formula and Midpoint Formula on the coordinate plane is derived from the Pthagorean Theorem a b c. In three dimensions, the coordinate grid contains the -ais and the -ais, as in two-dimensional geometr, and also a z-ais. An eample of a line segment drawn on a three-dimensional coordinate grid is shown at the right. The three-dimensional distance formula is much like the one for two dimensions. The distance from A (,, z ) to B (,, z ) can be found using d ( ) ( ) (z. z ) z (0,, 3) (, 3, 0) Eample Find the distance between the points (, 3, 0) and (0,, 3). d ( ) 03 ( ) 0 ( ) 3 Replace (,, z ) with (, 3, 0) and (,, z ) with (0,, 3). d or Simplif. Eercises. Find the distance between each pair of points A (, 3, ) and B (,, ). d. Find the distance between each pair of points C (, 3, ) and D (0,, 7). d 3. Find the distance between each pair of points E (,, 6) and F (, 3, ). d 33. Use what ou know about the midpoint formula for a segment graphed on a regular coordinate grid to make a conjecture about the formula for finding the coordinates of a midpoint in three-dimensions. z z Midpoint,,.. Find the midpoint for each segment in Eercises 3. (/, /, /), (/,, /), (3/,, ) Chapter 0 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Word Problem Practice Midpoint and Distance Formulas. EXHIBITS Museum planners want to place a statue directl in the center of their Special Ehibits Room. Suppose the room is placed on a coordinate plane as shown. What are the coordinates of the center of this room?. AIRPLANES A grid is superimposed on a map of Teas. Dallas has coordinates (00, ) and Amarillo has coordinates (00, 0). If each unit represents mile, how long will it take a plane fling at an average speed of 0 miles per hour to fl directl from Dallas to Amarillo? Round our answer to the nearest tenth of an hour. 0. hour (, ). WALKING Laura starts at the origin. She walks units to the right and then units up. How far awa from the origin is she? Round our answer to the nearest tenth.. units 3. SURVEILLANCE A grid is superimposed on a map of the area directl surrounding the home of a suspect. Detectives want to position themselves on opposite sides of the suspect s house. Coordinates are assigned to the suspect s home. Unit A is positioned at (, 6) on the coordinate plane. Where should Unit B be located so that the suspect s home is centered between the two units? Unit A TRAVEL For Eercises and 6, use the following information and the figure below. The Martinez famil is planning a trip from their home in Fort Lauderdale to Tallahassee. The plan to stop overnight at a location about halfwa between the two cities.. What are the coordinates of the point halfwa between Tallahassee and Fort Lauderdale? Which of the cities on the map is closest to this point? (0, ); Land Lakes Tallahassee Jacksonville Gainsville Datona Beach Land Lakes St. Petersburg Tampa 0 00 mi rlando Plant Cit Fort Lauderdale Miami (, 0) Suspect s home 6. How man miles is it from Fort Lauderdale to Tallahassee? Round our answer to the nearest mile. mi Chapter 0 0 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A Glencoe Algebra

5 Answers (Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Stud Guide and Intervention Parabolas Equations of Parabolas A parabola is a curve consisting of all points in the coordinate plane that are the same distance from a given point (the focus) and a given line (the directri). The following chart summarizes important information about parabolas. Standard Form of Equation a( h) k a( k) h Ais of Smmetr h k Verte (h, k) (h, k) Focus h, k a h a, k Directri k a h a Direction of pening upward if a 0, downward if a 0 right if a 0, left if a 0 Length of Latus Rectum units units a a Eample Identif the coordinates of the verte and focus, the equations of the ais of smmetr and directri, and the direction of opening of the parabola with equation. riginal equation ( 6) Factor from the -terms. ( 6 ) ( ) Complete the square on the right side. ( 6 ) () The added to complete the square is multiplied b. ( 3) 3 Write in standard form. 7 The verte of this parabola is located at (3, 3), the focus is located at 3,, the equation of the ais of smmetr is 3, and the equation of the directri is 3. The parabola opens upward. Eercises Identif the coordinates of the verte and focus, the equations of the ais of smmetr and directri, and the direction of opening of the parabola with the given equation (3, 3), (, ),, 7, (0, ),,, 3 3,, 3,,,, 0, 3, up down right Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- Lesson Reading Guide Chapter 0 Glencoe Algebra Write an equation of each parabola described below.. focus (, 3), directri. verte (, ), focus, 6( 3) 3( ) Chapter 0 3 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Parabolas Get Read for the Lesson Read the introduction to Lesson 0- in our tetbook. Name at least two reflective objects that might have the shape of a parabola. Sample answer: telescope mirror, satellite dish Read the Lesson. In the parabola shown in the graph, the point (, ) is called the verte and the point (, 0) is called the (, 0) focus. The line is called the directri, and the line is called the (, ) ais of smmetr. a. Write the standard form of the equation of a parabola that opens upward or downward. a( h) k b. The parabola opens downward if a 0 and opens upward if a 0. The equation of the ais of smmetr is h, and the coordinates of the verte are (h, k) 3. A parabola has equation ( ). This parabola opens to the left. It has verte (, ) and focus (, ). The directri is 6. The length of the latus rectum is units. Remember What You Learned. How can the wa in which ou plot points in a rectangular coordinate sstem help ou to remember what the sign of a tells ou about the direction in which a parabola opens? Sample answer: In plotting points, a positive -coordinate tells ou to move to the right and a negative -coordinate tells ou to move to the left. This is like a parabola whose equation is of the form ; it opens to the right if a 0 and to the left if a 0. Likewise, a positive -coordinate tells ou to move up and a negative -coordinate tells ou to move down. This is like a parabola whose equation is of the form ; it opens upward if a 0 and downward if a 0... Chapter 0 A Glencoe Algebra

6 Answers (Lesson 0-) Lesson 0- NAME DATE PERID 0- Skills Practice Parabolas Write each equation in standard form [ ()] ( ) 3 [ ()] (3) Identif the coordinates of the verte and focus, the equations of the ais of smmetr and directri, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola.. ( ). ( ) 3 6. ( 3) verte: (, 0); 3 verte: (3, ); verte: (3, ); focus:, ; focus: 3, ; focus: 3, 3 ; ais of smmetr: ais of smmetr: ais of smmetr: ; ; 3; directri: ; directri: ; directri: ; opens up; opens right; opens down; latus rectum: unit latus rectum: unit latus rectum: unit 3 Write an equation for each parabola described below. Then draw the graph. 7. verte (0, 0),. verte (, ),. verte (, 3), focus 0, focus, directri 3 ( ) ( 3) 7 Chapter 0 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Stud Guide and Intervention (continued) Parabolas Graph Parabolas To graph an equation for a parabola, first put the given equation in standard form. a( h) k for a parabola opening up or down, or a( k) h for a parabola opening to the left or right Use the values of a, h, and k to determine the verte, focus, ais of smmetr, and length of the latus rectum. The verte and the endpoints of the latus rectum give three points on the parabola. If ou need more points to plot an accurate graph, substitute values for points near the verte. Eample Graph 3 ( ). In the equation, a 3, h, k. The parabola opens up, since a 0. verte: (, ) ais of smmetr: 3 focus:, or, 3 length of latus rectum: 3 or 3 units endpoints of latus rectum:, 3,, 3 Eercises The coordinates of the focus and the equation of the directri of a parabola are given. Write an equation for each parabola and draw its graph.. (3, ),. (, ), 6 3. (, ), 3 ( 3) 3 ( ) ( ) Chapter 0 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A6 Glencoe Algebra

7 Answers (Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Word Problem Practice Parabolas. PRJECTILE A projectile follows the graph of the parabola 6. Sketch the path of the projectile b graphing the parabola. 6. TELESCPES An astronomer is working 3 with a large reflecting telescope. The reflecting mirror in the telescope has the parabolic cross section shown in the graph whose equation is given b ( ). Each unit represents meter. The astronomer is standing at the origin. How far from the focus of the parabola is the point on the mirror directl over the astronomer s head? 6. CMMUNICATIN David has just made a large parabolic dish whose cross section is based on the graph of the parabola 0.. Each unit represents one foot and the diameter of his dish is feet. He wants to make a listening device b placing a microphone at the focus of the parabola. Where should the microphone be placed? foot awa from the verte along the ais (at the point (0, ) with respect to the graph) 3. BRIDGES A bridge is in the shape of a parabola that opens downward. The equation of the parabola to model the arch of the bridge is given b 6 6, where each unit is equivalent to ard. The -ais is the ground level. What is the maimum height of the bridge above the ground? 6 d m BRIDGES For Eercises and 6, use the following information. Part of the Sdne Harbor Bridge in Sdne, Australia, can be modeled b a parabolic arch. If each unit corresponds to 0 meters, the arch would pass through the points at (, ), (0, 0), and (, ).. Write the equation of the parabola to model the arch. 0 Cross section of mirror surface 6. Identif the coordinates of the focus of this parabola. (0,.) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- Practice Chapter 0 6 Glencoe Algebra Chapter 0 7 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Parabolas Write each equation in standard form ( 3) [ (3)] () 3[ ()] Identif the coordinates of the verte and focus, the equations of the ais of smmetr and directri, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola.. ( ) ( ) 3 verte: (, 3); verte: (, 0); verte: (3, ); focus:, 3 ; focus:,0; focus:, ; ais: ; ais: 0; ais: ; directri: ; directri: ; directri: 3 ; opens up; opens left; opens right; latus rectum: unit latus rectum: 3 units latus rectum: unit Write an equation for each parabola described below. Then draw the graph. 7. verte (0, ),. verte (, ),. verte (, 3), 7 focus 0, 3 directri 3 ais of smmetr, latus rectum: units, a 0 ( ) ( ) 3 0. TELEVISIN Write the equation in the form a for a satellite dish. Assume that the bottom of the upward-facing dish passes through (0, 0) and that the distance from the bottom to the focus point is inches. 3 Chapter 0 A7 Glencoe Algebra

8 Answers (Lesson 0-) Lesson 0-0- NAME DATE PERID Spreadsheet Activit Parabolas You have learned man of the characteristics of parabolas with vertical and horizontal aes of smmetr. The information is summarized in the table at the right. You can use what ou know to create a spreadsheet to analze given equations of parabolas. form of equation a( h) k a( k) h verte (h, k) (h, k) ais of smmetr h k focus ( h, k ) h ( h ) a a, k directri k a h direction of opening upward if a 0, right if a 0, left downward if a 0 if a 0 length of latus a units a units rectum a Eercises The spreadsheet below uses the equation of a parabola in the form a( h) k or a( k) h to find information about the parabola. or is entered in Column D and the values of a, h, and k are entered into Columns A, B, and C respectivel. 3 A B C D E a h k or Sheet Sheet Sheet 3 Verte Length of Latus Rectum (units) 3,, 3, F G H I J Ais of Smmetr 3 Focus,,.0 3,.7 Directri.. Direction of pening Right Upward Downward. Which row represents the equation 3 0? row 3. Write the standard form of the equation represented b row. ( ) 3 3. What formula should be used in cell F? /ABS(A). Find the verte, length of latus rectum, ais of smmetr, focus, directri, and direction of opening of a parabola with equation ( ) ( ). (, ); ; ; (7, ); ; left Chapter 0 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Enrichment Limits Sequences of numbers with a rational epression for the general term often approach some number as a finite limit. For eample, the reciprocals of the positive integers approach 0 as n gets larger and larger. This is written using the notation shown below. The smbol stands for infinit and n means that n is getting larger and larger, or n goes to infinit.,, 3,,,, lim n n n 0 Eample Find lim n n (n ) It is not immediatel apparent whether the sequence approaches a limit or not. But notice what happens if we divide the numerator and denominator of the general term b n. n n (n ) n n n n n n nn n n n The two fractions in the denominator will approach a limit of 0 as n gets ver large, so the entire epression approaches a limit of. Eercises Find the following limits.. lim n n n 3 n 0. lim 6 n n n 0 3. lim (n n ) n. lim n n 3n 3 Chapter 0 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A Glencoe Algebra

9 Answers (Lesson 0-3) Lesson 0-3 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0-3 NAME DATE PERID Stud Guide and Intervention Circles Equations of Circles The equation of a circle with center (h, k) and radius r units is ( h) ( k) r. Eample Write an equation for a circle if the endpoints of a diameter are at (, ) and (6, 3). Use the midpoint formula to find the center of the circle. (h, k), Midpoint formula 6 (3), (, ) (, ), (, ) (6, 3), or (, ) Simplif. Use the coordinates of the center and one endpoint of the diameter to find the radius. r ( ( ) ) Distance formula r ( ) ) ( (, ) (, ), (, ) (, ) () Simplif. The radius of the circle is, so r. An equation of the circle is ( ) ( ). Eercises Write an equation for the circle that satisfies each set of conditions.. center (, 3), radius 6 ( ) ( 3) 36. center (, 6), radius ( ) ( 6) 6 3. center (, ), passes through (, 6) ( ) ( ) 3. endpoints of a diameter at (6, 6) and (0, ) ( ) ( ) 3. center (3, 6), tangent to the -ais ( 3) ( 6) center (, 7), tangent to ( ) ( 7) center at (, ), tangent to ( ) ( ) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0-3 Chapter 0 0 Glencoe Algebra. center (7, 7), passes through (, ) ( 7) ( 7). endpoints of a diameter are (, ) and (, ) ( ) ( ) 0. endpoints of a diameter are (, 3) and (6, ) ( ) (.). Chapter 0 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Lesson Reading Guide Circles Get Read for the Lesson Read the introduction to Lesson 0-3 in our tetbook. A large home improvement chain is planning to enter a new metropolitan area and needs to select locations for its stores. Market research has shown that potential customers are willing to travel up to miles to shop at one of their stores. How can circles help the managers decide where to place their store? Sample answer: A store will draw customers who live inside a circle with center at the store and a radius of miles. The management should select locations for which as man people as possible live within a circle of radius miles around one of the stores. Read the Lesson. a. Write the equation of the circle with center (h, k) and radius r. ( h) ( k) r b. Write the equation of the circle with center (, 3) and radius. ( ) ( 3) c. The circle with equation ( ) has center (, 0) and radius d. The circle with equation ( 0) ( 0) has center (0, 0) and radius.. a. In order to find center and radius of the circle with equation 6 3 0, it is necessar to complete the square. Fill in the missing parts of this process ( ) ( 3 ) 6 b. This circle has radius and center at (, 3). Remember What You Learned 3. How can the distance formula help ou to remember the equation of a circle? Sample answer: Write the distance formula. Replace (, ) with (h, k) and (, ) with (, ). Replace d with r. Square both sides. Now ou have the equation of a circle.. Chapter 0 A Glencoe Algebra

10 Answers (Lesson 0-3) Lesson 0-3 NAME DATE PERID 0-3 Skills Practice Circles Write an equation for the circle that satisfies each set of conditions.. center (0, ), radius unit. center (, ), radius units ( ) ( ) ( ) 6 3. center (, 0), radius units. center (, ), radius 3 units ( ) ( ) ( ). center (, ), radius units 6. center (6, ), radius units ( ) ( ) 6 ( 6) ( ) 7. endpoints of a diameter at (, 0) and (, 0). endpoints of a diameter at (, 0) and (, 6) ( ) ( 3). center at (7, 3), passes through the origin ( 7) ( 3) 0. center at (, ), passes through (, ) ( ) ( ). center at (6, ), tangent to -ais ( 6) ( ) 36. center at (, ), tangent to -ais ( ) ( ) Find the center and radius of the circle with the given equation. Then graph the circle. 3.. ( ) ( ). ( ) 6 (0, 0), 3 units (, ), units (, 0), units 6. ( 3) 7. ( ) ( ). 3 0 (0, 3), units (, ), 7 units (0, ), 6 units Chapter 0 3 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0-3 NAME DATE PERID Stud Guide and Intervention (continued) Circles Graph Circles To graph a circle, write the given equation in the standard form of the equation of a circle, ( h) ( k) r. Plot the center (h, k) of the circle. Then use r to calculate and plot the four points (h r, k), (h r, k), (h, k r), and (h, k r), which are all points on the circle. Sketch the circle that goes through those four points. Eample Find the center and radius of the circle whose equation is. Then graph the circle. ( ) ( ) 6 Therefore, the circle has its center at (, ) and a radius of 6. Four points on the circle are (3, ), (, ), (, ), and (, 6). Eercises Find the center and radius of the circle with the given equation. Then graph the circle.. ( 3). ( ) 3. ( ) ( 3) (3, 0), r 3 (0, ), r (, 3), r 3. ( ) ( ) (, ), r (, ), r (, 3), r Chapter 0 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A0 Glencoe Algebra

11 Answers (Lesson 0-3) Lesson 0-3 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0-3 NAME DATE PERID Word Problem Practice Circles. RADAR A scout plane is equipped with radar. The boundar of the radar s range is given b the circle ( ) ( 6) 00. Each unit corresponds to one mile. What is the maimum distance that an object can be from the plane and still be detected b its radar? Scout plane. PLS The pool on an architectural floor plan is given b the equation 6 0. What point on the edge of the pool is farthest from the origin? (6, ) TREASURE For Eercises and 6, use the following information. A mathematicall inclined pirate decided to hide the location of a treasure b marking it as the center of a circle given b an equation in non-standard form. 70 mi. STRAGE An engineer uses a coordinate plane to show the laout of a side view of a storage building. The -ais represents a wall and the -ais represents the floor. A 0-meter diameter clinder rests on its side flush against the wall. n the side view, the clinder is represented b a circle in the first quadrant that is tangent to both aes. Each unit represents meter. What is the equation of this circle? ( ) ( ) 3. FERRIS WHEEL The Teas Star, the largest Ferris wheel in North America, is located in Dallas, Teas. It weighs 67, pounds and can hold 6 riders in its gondolas. The Teas Star has a diameter of feet. Use the rectangular coordinate sstem with the origin on the ground directl below the center of the wheel and write the equation of the circle that models the Teas Star. ( 06),36 The secret circle can be represented b: 0.. Rewrite the equation of the circle in standard form. ( ) ( 7) 6. Draw the circle on the map. Where is the treasure? See circle on map at (, 7); the southwest corner of Meadow Madness. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0-3 Chapter 0 Glencoe Algebra Chapter 0 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Practice Circles Write an equation for the circle that satisfies each set of conditions.. center (, ), radius units. center (0, 0), radius units ( ) ( ) center, 3, radius units. center (.,.), radius 0. unit ( 3) 0 (.) (.) 0.. endpoints of a diameter at (, ) and (0, ) ( ) ( 7) 6. center at (, ), passes through (, ) ( ) ( ) 7 7. center at (6, ), tangent to -ais ( 6) ( ) Find the center and radius of the circle with the given equation. Then graph the circle.. ( 3) (3, 0), units (0, 0), units (, 3), 6 units. ( ) (, ), units (3, 0), 3 units (, 3), 3 units WEATHER For Eercises and, use the following information. n average, the circular ee of a hurricane is about miles in diameter. Gale winds can affect an area up to 300 miles from the storm s center. In 00, Hurricane Katrina devastated southern Louisiana. A satellite photo of Katrina s landfall showed the center of its ee on one coordinate sstem could be approimated b the point (0, 6).. Write an equation to represent a possible boundar of Katrina s ee. ( 0) ( 6) 6.. Write an equation to represent a possible boundar of the area affected b gale winds. ( 0) ( 6) 0,000 Chapter 0 A Glencoe Algebra

12 Answers (Lesson 0-3) Lesson NAME DATE PERID Graphing Calculator Activit Matrices and Equations of Circles A graphing calculator can be used to write the equation of a circle in the form D E F 0 given an three points on the circle. Eample radius of each circle. Write the equation of the circle that passes through the given points. Identif the center and a. A(, 3), B(, ), and C(, ) Substitute each ordered pair for (, ) in D E F 0 to form the a sstem of equations. D 3E F 3 D E F D E F 6 Solve the sstem using a matri equation to find D, E,and F. Replace the coefficients in the epanded form. Then, complete the square to write the equation in standard form to identif the center and radius. Kestrokes: nd [MATRX] ENTER 3 ENTER 3 ENTER ENTER 3 ENTER ENTER ( ) ENTER ENTER ENTER ( ) ENTER ( ) ENTER ENTER nd [MATRX] [EDIT] ENTER 3 ENTER ENTER ( ) 3 ENTER ( ) ENTER ( ) 6 ENTER nd [QUIT] nd [MATRX] ENTER nd [MATRX] ENTER Thus, D, E, and F 0. The epanded form is 0 0. After completing the square, the standard form is ( ) ( ). The center is (, ), and the radius is. b. A(, 3), B(6, ), and C(0, 7) Find a sstem of equations. Then enter the equations into an augmented matri. Reduce the matri to row reduced echelon form using the rref( command. The row reduced echelon form of an augmented matri will displa the solution to the sstem. D 3E + F 3 6D E F 6 7E F Kestrokes: Enter the sstem of equations as [A], a 3 augmented matri. Then use the reduced row echelon form b pressing nd [MATRX] ALPHA [B] nd [MATRX] ENTER ENTER. The solution is D = 0, E =, and F =. The epanded form is 0 = 0, standard form is ( ) ( ) 0. The center is (, ) and the radius is. Eercises Write the equation of the circle that passes through the given points. Identif the center and radius of each circle.. (0, ), (3, ), and (6, ). (7, ), (, ), and (3, ) 3. (, 7), (, 0), and (0, ) ; 0 0; ; C(3, 3), R C(7, ), R C(3, ), R 3 Chapter 0 7 Glencoe Algebra. ) Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0-3 NAME DATE PERID Enrichment Tangents to Circles A line that intersects a circle in eactl one point is a tangent to the circle. In the diagram, line is tangent to the circle with equation at the point whose coordinates are (3, ). (3, ) A line is tangent to a circle at a point P on the circle if and onl if the line is perpendicular to the radius from the center of the circle to point P. This fact enables ou to find an equation of the tangent to a circle at a point P if ou know an equation for the circle and the coordinates of P. Eercises Use the diagram above to solve each problem.. What is the slope of the radius to the point with coordinates (3, )? What is the slope of the tangent to that point? 3, 3. Find an equation of the line that is tangent to the circle at (3, ) If k is a real number between and, how man points on the circle have -coordinate k? State the coordinates of these points in terms of k. two, (k, k ). Describe how ou can find equations for the tangents to the points ou named for Eercise 3. Use the coordinates of (0, 0) and of one of the given points. Find the slope of the radius to that point. Use the slope of the radius to find what the slope of the tangent must be. Use the slope of the tangent and the coordinates of the point on the circle to find an equation for the tangent.. Find an equation for the tangent at (3, ). 3 Chapter 0 6 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A Glencoe Algebra

13 Answers (Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Stud Guide and Intervention Ellipses Equations of Ellipses An ellipse is the set of all points in a plane such that the sum of the distances from two given points in the plane, called the foci, is constant. An ellipse has two aes of smmetr which contain the major and minor aes. In the table, the lengths a, b, and c are related b the formula c a b. Standard Form of Equation Center (h, k) (h, k) Direction of Major Ais Horizontal Vertical Foci (h c, k), (h c, k) (h, k c), (h, k c) Length of Major Ais a units a units Length of Minor Ais b units b units Write an equation for the ellipse shown. The length of the major ais is the distance between (, ) and (, ). This distance is 0 units. a 0, so a The foci are located at (, 6) and (, 0), so c 3. b a c 6 The center of the ellipse is at (, 3), so h, k 3, a, and b 6. The major ais is vertical. An equation of the ellipse is. Write an equation for the ellipse that satisfies each set of conditions.. endpoints of major ais at (7, ) and (, ), endpoints of minor ais at (, 0) and (, ) ( ) 36. major ais units long and parallel to the -ais, minor ais units long, center at (, ) ( ) 6 ( ) 3. endpoints of major ais at (, ) and (, ), foci at (3, ) and (, ) ( ) ( ) 36 3 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- Chapter 0 Glencoe Algebra ( h) a ( k) b ( k) a ( h) b Eample Eercises. endpoints of major ais at (3, ) and (3, ), endpoints of minor ais at (, 6) and (7, 6) ( 6) ( 3) 6 6. minor ais 6 units long and parallel to the -ais, major ais units long, center at (6, ) ( ) 36 ( ) ( 6) ( 3) ( ) 6 F F Chapter 0 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Lesson Reading Guide Ellipses Get Read for the Lesson Read the introduction to Lesson 0- in our tetbook. Is the Earth alwas the same distance from the Sun? Eplain our answer using the words circle and ellipse. No; if the Earth s orbit were a circle, it would alwas be the same distance from the Sun because ever point on a circle is the same distance from the center. However, the Earth s orbit is an ellipse, and the points on an ellipse are not all the same distance from the center. Read the Lesson. An ellipse is the set of all points in a plane such that the sum of the distances from two fied points is constant. The two fied points are called the foci of the ellipse.. Consider the ellipse with equation. a. For this equation, a 3 and b. b. Write an equation that relates the values of a, b, and c. c a b c. Find the value of c for this ellipse. 3. Consider the ellipses with equations and. Complete the following table to describe characteristics of their graphs. 6 Standard Form of Equation 6 Direction of Major Ais vertical horizontal Direction of Minor Ais horizontal vertical Foci (0, 3), (0, 3) (, 0), (, 0) Length of Major Ais 0 units 6 units Length of Minor Ais units units Remember What You Learned. Some students have trouble remembering the two standard forms for the equation of an ellipse. How can ou remember which term comes first and where to place a and b in these equations? The -ais is horizontal. If the major ais is horizontal, the a first term is. The -ais is vertical. If the major ais is vertical, the first term is. a is alwas the larger of the numbers a and b. a Chapter 0 A3 Glencoe Algebra

14 Answers (Lesson 0-) Lesson 0- NAME DATE PERID 0- Skills Practice Ellipses Write an equation for each ellipse... (0, ) 3. (0, ) (0, ) ( 3, 0) (0, ) (0, 3) (, ) (, ) (3, 0) (0, ) (0, 3) (0, ) 6 6 ( ) Write an equation for the ellipse that satisfies each set of conditions.. endpoints of major ais. endpoints of major ais 6. endpoints of major ais at (0, 6) and (0, 6), at (, 6) and (, 6), at (7, 3) and (7, ), endpoints of minor ais endpoints of minor ais endpoints of minor ais at (3, 0) and (3, 0) at (, ) and (, ) at (, 6) and (, 6) 36 ( ) ( 6) ( 6) ( 7) 7. major ais units long. endpoints of major ais. endpoints of major ais at and parallel to -ais, at (6, 0) and (6, 0), foci (0, ) and (0, ), foci at minor ais units long, at (3, 0) and (3, 0) (0, 3) and (0, 3) center at (0, 0) Find the coordinates of the center and foci and the lengths of the major and minor aes for the ellipse with the given equation. Then graph the ellipse (0, 0); (0, ); (0, 0); (6, 0); (0, 0), (0, 6); 0; ; 6 ; 0 Chapter 0 3 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Stud Guide and Intervention (continued) Ellipses Graph Ellipses To graph an ellipse, if necessar, write the given equation in the standard form of an equation for an ellipse. ( h) a ( k) (for ellipse with major ais horizontal) or b ( k) a ( h) (for ellipse with major ais vertical) b Use the center (h, k) and the endpoints of the aes to plot four points of the ellipse. To make a more accurate graph, use a calculator to find some approimate values for and that satisf the equation. Eample Graph the ellipse ( ) 6( 6 ) 3 ( ) 6( 6 ) 3 ( ) 6( 3) ( ) ( 3) 6 The center of the ellipse is (, 3). Since a 6, a 6. Since b, b. The length of the major ais is 6, and the length of the minor ais is. Since the -term has the greater denominator, the major ais is horizontal. Plot the endpoints of the aes. Then graph the ellipse Eercises Find the coordinates of the center and the lengths of the major and minor aes for the ellipse with the given equation. Then graph the ellipse.. (0, 0), 3,6. (0, 0), 0, 3. 3 (0, 3),, (, ), 6, 6 Chapter 0 30 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A Glencoe Algebra

15 Answers (Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Word Problem Practice Ellipses. PERSPECTIVE A graphic designer uses an ellipse to draw a circle from the horizontal perspective. The equation used is. Graph this ellipse.. ASTRNMY The orbit of an asteroid is given b the equation, 00 where each unit represents one astronomical unit (i.e. the distance from Sun to Earth). What are the lengths of the major and minor aes of the orbit? Major ais: astronomical units, Minor ais: 0 astronomical units. ECHES The walls of an elliptical room are given b the equation. Two people want to stand at the foci of the ellipse so that the can whisper to each other without anbod else hearing. What are the coordinates of the foci? (3, 0) and (3, 0) 3. FLASHLIGHTS Daniella ended up doing her math homework late at night. To avoid disturbing others, she worked in bed with a pen light. ne problem asked her to draw an ellipse. She noticed that her pen light created an elliptical patch of light on her paper, so she simpl traced the outline of the patch of light. The outline of the ellipse is shown below. What is the equation of this ellipse in standard form? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- Practice Chapter 0 3 Glencoe Algebra 6 MDELING For Eercises and 6, use the following information. James wants to tr to make an ellipse using a piece of string 6 inches long. He tacks the two ends down 0 inches apart. He then takes a pen and pulls the string taut. He keeps the string taut and pulls the pen around the tacks. B doing this, he creates an ellipse.. Determine the lengths of the major and minor aes of the ellipse that James drew. Major ais has length 6, minor ais has length. 6. If a coordinate grid is overlaid on the ellipse so that the tacks are located at (, 0) and (, 0), what is the equation of the ellipse in standard form? 6 36 Chapter 0 33 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Ellipses Write an equation for each ellipse.. (0, 3). (0, ) (0, ) 3. ( 6, 3) (, 3) (, 0) (, 0) 6 6 (0, 3) (, 3) (3, 3) (0, ) (0, ) ( ) ( ) ( 3) Write an equation for the ellipse that satisfies each set of conditions.. endpoints of major ais. endpoints of major ais 6. major ais 0 units long at (, 0) and (, 0), at (, ) and (, ), and parallel to -ais, endpoints of minor ais endpoints of minor ais minor ais 0 units long, at (0, 3) and (0, 3) at (, 3) and (7, 3) center at (, ) ( 3) ( ) ( ) 00 ( ) 7. major ais 0 units long,. major ais 6 units long,. endpoints of minor ais minor ais 6 units long center at (0, 0), foci at at (0, ) and (0, ), foci and parallel to -ais, (0, ) and (0, ) at (, 0) and (, 0) center at (, ) ( ) ( ) 6 0 Find the coordinates of the center and foci and the lengths of the major and minor aes for the ellipse with the given equation. Then graph the ellipse (0, 0); (0, 7); ; 6 (3, ); (3, 3 ); (, 3); ; ( 6, 3); ; 0 ( ) 36 ( 3) ( ) ( 3) 3. SPRTS An ice skater traces two congruent ellipses to form a figure eight. Assume that the center of the first loop is at the origin, with the second loop to its right. Write an equation to model the first loop if its major ais (along the -ais) is feet long and its minor ais is 6 feet long. Write another equation to model the second loop. 36 ( ) 36 ; Chapter 0 A Glencoe Algebra

16 Answers (Lessons 0- and 0-) Lesson 0-0- NAME DATE PERID Lesson Reading Guide Hperbolas Get Read for the Lesson Read the introduction to Lesson 0- in our tetbook. Look at the sketch of a hperbola in the introduction to this lesson. List three was in which hperbolas are different from parabolas. Sample answer: A hperbola has two branches, while a parabola is one continuous curve. A hperbola has two foci, while a parabola has one focus. A hperbola has two vertices, while a parabola has one verte. Read the Lesson. The graph at the right shows the hperbola whose equation in standard form is The point (0, 0) is the center of the hperbola. (, 0) (, 0) (, 0) (, 0) The points (, 0) and (, 0) are the vertices of the hperbola. The points (, 0) and (, 0) are the foci of the hperbola. The segment connecting (, 0) and (, 0) is called the transverse ais. The segment connecting (0, 3) and (0, 3) is called the conjugate ais. 3 3 The lines and are called the asmptotes.. Stud the hperbola graphed at the right. The center is (0, 0). The value of a is. The value of c is. To find b, solve the equation c a b. The equation in standard form for this hperbola is. Remember What You Learned 3. What is an eas wa to remember the equation relating the values of a, b, and c for a hperbola? This equation looks just like the Pthagorean Theorem, although the variables represent different lengths in a hperbola than in a right triangle. Chapter 0 3 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Enrichment Eccentricit In an ellipse, the ratio a c is called the eccentricit and is denoted b the letter e. Eccentricit measures the elongation of an ellipse. The closer e is to 0, the more an ellipse looks like a circle. The closer e is to, the more elongated it is. Recall that the equation of an ellipse is a b or b a where a is the length of the major ais, and that c a b. Find the eccentricit of each ellipse rounded to the nearest hundredth Is a circle an ellipse? Eplain our reasoning. Yes; it is an ellipse with eccentricit 0.. The center of the sun is one focus of Earth's orbit around the sun. The length of the major ais is 6,000,000 miles, and the foci are 3,00,000 miles apart. Find the eccentricit of Earth's orbit. approimatel An artificial satellite orbiting the earth travels at an altitude that varies between 3 miles and 3 miles above the surface of the earth. If the center of the earth is one focus of its elliptical orbit and the radius of the earth is 30 miles, what is the eccentricit of the orbit? approimatel 0.0 Chapter 0 3 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A6 Glencoe Algebra

17 Answers (Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Stud Guide and Intervention (continued) Hperbolas Graph Hperbolas To graph a hperbola, write the given equation in the standard form of an equation for a hperbola ( h) a ( k) a ( k) if the branches of the hperbola open left and right, or b ( h) b if the branches of the hperbola open up and down Graph the point (h, k), which is the center of the hperbola. Draw a rectangle with dimensions a and b and center (h, k). If the hperbola opens left and right, the vertices are (h a, k) and (h a, k). If the hperbola opens up and down, the vertices are (h, k a) and (h, k a). Eample Draw the graph of Complete the squares to get the equation in standard form ( 6 ) ( ) 6 6( 6 ) ( ) 6 0 6( 3) ( ) ( 3) ( ) 6 The center of the hperbola is (, 3). According to the equation, a and b 6, so a and b 6. The transverse ais is vertical, so the vertices are (, ) and (, ). Draw a rectangle with vertical dimension and horizontal dimension 6.. The diagonals of this rectangle are the asmptotes. The branches of the hperbola open up and down. Use the vertices and the asmptotes to sketch the hperbola. Eercises Find the coordinates of the vertices and foci and the equations of the asmptotes for the hperbola with the given equation. Then graph the hperbola. 6 ( ) ( 3) 3. (, 0), (, 0); (, ), (, ); (0, ), (0, ); (, 0), (, 0); (, 3 0 ), (0, ), (0, ); (, 3 0 ); 3 3, Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- Chapter 0 36 Glencoe Algebra Chapter 0 37 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Stud Guide and Intervention Hperbolas Equations of Hperbolas A hperbola is the set of all points in a plane such that the absolute value of the difference of the distances from an point on the hperbola to an two given points in the plane, called the foci, is constant. In the table, the lengths a, b, and c are related b the formula c a b. ( h) a ( k) b ( k) a ( h) b Standard Form of Equation b a Equations of the Asmptotes k a ( h) k b ( h) Transverse Ais Horizontal Vertical Foci (h c, k), (h c, k) (h, k c), (h, k c) Vertices (h a, k), (h a, k) (h, k a), (h, k a) Eample Write an equation for the hperbola with vertices (, ) and (6, ) and foci (, ) and (, ). Use a sketch to orient the hperbola correctl. The center of the hperbola is the midpoint of the segment joining the two 6 vertices. The center is (, ), or (, ). The value of a is the distance from the center to a verte, so a. The value of c is the distance from the center to a focus, so c 6. c a b 6 b b Use h, k, a, and b to write an equation of the hperbola. ( ) ( ) 6 0 Eercises Write an equation for the hperbola that satisfies each set of conditions.. vertices (7, 0) and (7, 0), conjugate ais of length 0 ( ) ( 3) 7. vertices (, 3) and (, 3), foci (, 3) and (7, 3) ( ) 6 ( ) 3. vertices (, 3) and (, ), conjugate ais of length. vertices (, 0) and (, 0), equation of asmptotes ( ) 6 ( ). vertices (, 6) and (, ), foci (, 0) and (, 6) Chapter 0 A7 Glencoe Algebra

18 Answers (Lesson 0-) Lesson 0- NAME DATE PERID 0- Practice Hperbolas Write an equation for each hperbola (0, 3) ( 3, 3) (0, 3) (0, 3) (0, 3) ( 3, ) ( 3, ) ( 3, 3) (, ) (, ) (3, ) 36 ( ) Write an equation for the hperbola that satisfies each set of conditions.. vertices (0, 7) and (0, 7), conjugate ais of length units. vertices (, ) and (, ), conjugate ais of length 6 units ( ) 6. vertices (, 0) and (, 0), foci (6, 0) ( ) 7. vertices (, ) and (, 3), foci (, ) ( 3) ( ) ( ) 6 ( ) 6 ( ) Find the coordinates of the vertices and foci and the equations of the asmptotes for the hperbola with the given equation. Then graph the hperbola. 6 ( ) ( ) ( ) ( 3).. 0. (0, ); (0, ); (, 3), (, ); (3, 0), (3, ); (, ); (3, ); ( ) ( 3). ASTRNMY Astronomers use special X-ra telescopes to observe the sources of celestial X ras. Some X-ra telescopes are fitted with a metal mirror in the shape of a hperbola, which reflects the X ras to a focus. Suppose the vertices of such a mirror are located at (3, 0) and (3, 0), and one focus is located at (, 0). Write an equation that models the hperbola formed b the mirror. 6 Chapter 0 3 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID 0- Skills Practice Hperbolas Write an equation for each hperbola (0, 6) (0, 6) (, 0) (, 0) (, 0) (, 0) (0, 6) (0, 6) (, 0) (, 0) (, 0) (, 0) 6 36 Write an equation for the hperbola that satisfies each set of conditions.. vertices (, 0) and (, 0), conjugate ais of length 6 6. vertices (0, 6) and (0, 6), conjugate ais of length vertices (0, 3) and (0, 3), conjugate ais of length 0 7. vertices (, 0) and (, 0), conjugate ais of length. vertices (3, 0) and (3, 0), foci (, 0) 6. vertices (0, ) and (0, ), foci (0, 3) ( 3) ( ) 0. vertices (0, ) and (6, ), foci (3 3, ) Find the coordinates of the vertices and foci and the equations of the asmptotes for the hperbola with the given equation. Then graph the hperbola (3, 0); (3, 0); (0, 7); (0, ); (, 0); (7, 0); 7 3 Chapter 0 3 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A Glencoe Algebra

19 Answers (Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Enrichment Rectangular Hperbolas A rectangular hperbola is a hperbola with perpendicular asmptotes. For eample, the graph of is a rectangular hperbola. A hperbola with asmptotes that are not perpendicular is called a nonrectangular hperbola. The graphs of equations of the form c, where c is a constant, are rectangular hperbolas. Make a table of values and plot points to graph each rectangular hperbola below. Be sure to consider negative values for the variables. See students tables Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- Chapter 0 0 Glencoe Algebra. Make a conjecture about the asmptotes of rectangular hperbolas. The coordinate aes are the asmptotes. Chapter 0 Glencoe Algebra Answers Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE PERID Word Problem Practice Hperbolas. LIGHTHUSES The location of a lighthouse is represented b the origin of a coordinate plane. A boat in the distance appears to be on a collision course with the lighthouse. However, the boat veers off and turns awa at the last moment, avoiding the rock shallows. The path followed b the boat is modeled b a branch of the hperbola with equation. If the unit length corresponds to a ard, how close did the boat come to the lighthouse? 30 d. FIND THE ERRR Curtis was tring to write the equation for a hperbola with a vertical transverse ais of length 0 and conjugate ais of length 6. The equation he got was. Did he make a mistake? If so, what did he do wrong? Yes. Curtis echanged the transverse and conjugate aes. 3. MIRRR At a carnival, designers are planning a funhouse. The plan to put a large hperbolic mirror inside this funhouse. The design the mirror s hperbolic cross section on graph paper using a hperbola with a horizontal transverse ais. The asmptotes are to be and so the mirror is somewhat shallow. The also want the vertices to be unit from the origin. What equation should the use for the hperbola? ASTRNMY Astronomers discover a new comet. The stud its path and discover that it can be modeled b a branch of a hperbola with equation 0 0. Rewrite this equation in standard form and find the center of the hperbola. ( ), center at (, 0) LIGHTNING For Eercises 7, use the following information. Brittan and Kirk were talking on the phone when Brittan heard the thunder from a lightning bolt outside. Eight seconds later, she could hear the same thunder over the phone. Brittan and Kirk live miles apart and sounds travels about mile ever seconds.. n a coordinate plane, assume that Brittan is located at (, 0) and Kirk is located at (, 0). Write an equation using the Distance Formula that describes the possible locations of the lightning strike. ( ) ( ) 6. Rewrite the equation ou wrote for Eercise so it is in the standard form for a hperbola Which branch of the hperbola corresponds to the places where the lightning bolt might have struck? the left branch, the branch with negative coordinates Chapter 0 A Glencoe Algebra

20 Answers (Lesson 0-6) Lesson NAME DATE PERID Stud Guide and Intervention Conic Sections Standard Form An conic section in the coordinate plane can be described b an equation of the form A B C D E F 0, where A, B, and C are not all zero. ne wa to tell what kind of conic section an equation represents is to rearrange terms and complete the square, if necessar, to get one of the standard forms from an earlier lesson. This method is especiall useful if ou are going to graph the equation. Eample Write the equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hperbola riginal equation 3 30 Isolate terms. 3( 0 ) ( ) Factor out common multiples. 3( 0 ) ( ) 3() ()() Complete the squares. 3( ) ( ) Simplif. ( ) ( ) 3 Divide each side b. The graph of the equation is a hperbola with its center at (, ). The length of the transverse ais is units and the length of the conjugate ais is 3 units. Eercises Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hperbola ( 3) 6 ( ) 3 ( 3) ( ) 0; circle ; ellipse ( ) ; parabola ( ) ( 7) ; circle ( ) ( ) ; hperbola 3( ) ; parabola ( ) 6 ( 3) 6 ( ) 6 ( ) ; hperbola ; ellipse ( ) ( ) ( 6) ; parabola ; ellipse ( ) ( 3) 6 ; hperbola ( ) ( ) ; circle Chapter 0 3 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0-6 NAME DATE PERID Lesson Reading Guide Conic Sections Get Read for the Lesson Read the introduction to Lesson 0-6 in our tetbook. The figures in the introduction show how a plane can slice a double cone to form the conic sections. Name the conic section that is formed if the plane slices the double cone in each of the following was: The plane is parallel to the base of the double cone and slices through one of the cones that form the double cone. circle The plane is perpendicular to the base of the double cone and slices through both of the cones that form the double cone. hperbola Read the Lesson. Name the conic section that is the graph of each of the following equations. Give the coordinates of the verte if the conic section is a parabola and of the center if it is a circle, an ellipse, or a hperbola. ( 3) 36 ( ) a. ellipse; (3, ) b. ( ) 7 parabola; (7, ) c. ( ) ( ) hperbola; (, ) d. ( 6) ( ) circle; (6, ). Each of the following is the equation of a conic section. For each equation, identif the values of A and C. Then, without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hperbola. a. 6 0 A ; C ; tpe of graph: ellipse b. 3 0 A ; C 0 ; tpe of graph: parabola c A ; C ; tpe of graph: circle d. 0 A ; C ; tpe of graph: hperbola Remember What You Learned 3. What is an eas wa to recognize that an equation represents a parabola rather than one of the other conic sections? If the equation has an term and term but no term, then the graph is a parabola. Likewise, if the equation has a term and term but no term, then the graph is a parabola. Chapter 0 Glencoe Algebra Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 0 A0 Glencoe Algebra