1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered


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1 Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter, we present several new tpes of graphs, called conic sections. These include circles, parabolas, ellipses, and hperbolas. These shapes are found in a variet of applications. For eample, a reflecting telescope has a mirror whose cross section is in the shape of a parabola, and planetar orbits are modeled b ellipses. As ou work through the chapter, ou will encounter a variet of equations associated with the conic sections. Match the equation with its description, and then use the letter net to each answer to complete the puzzle.. a. standard form of a parabola with a b horizontal ais of smmetr. r o. standard form of an ellipse centered at the origin. a k h f. standard form of a circle centered at (h, k). d t. standard form of a hperbola with horizontal transverse ais. a h k s. standard form of a circle centered at the origin. h k r c. standard form of a parabola with 7. a b r. vertical ais of smmetr distance formula He became a math teacher due to some prime 7 78
2 78 Chapter Conic Sections Section. Concepts. Distance Formula. Circles. Writing an Equation of a Circle Distance Formula and Circles. Distance Formula Suppose we are given two points, and, in a rectangular coordinate sstem. The distance between the two points can be found b using the Pthagorean theorem (Figure ). First draw a right triangle with the distance d as the hpotenuse. The length of the (, ) horizontal leg a is 0 0, and the length of the vertical leg b is 0 0. From the d b 0 0 Pthagorean theorem we have (, ) a 0 0 Figure  d a b Pthagorean theorem d Because distance is positive, reject the negative value. The Distance Formula The distance d between the points (, ) and (, ) is d Eample Finding the Distance Between Two Points Find the distance between the points, and, (Figure ). Solution:, and,, and, d Label the points. Appl the distance formula. (, ) d 7. (, ) Skill Practice Answers. Skill Practice Figure . Find the distance between the points (, ) and (, ).
3 Section. Distance Formula and Circles 787 TIP: The order in which the points are labeled does not affect the result of the distance formula. For eample, if the points in Eample had been labeled in reverse, the distance formula would still ield the same result:, and, d, and,. Circles A circle is defined as the set of all points in a plane that are equidistant from a fied point called the center. The fied distance from the center is called the radius and is denoted b r, where r 7 0. Suppose a circle is centered at the point (h, k) and has radius, r (Figure ). The distance formula can be used to derive an equation of the circle. Let (, ) be an arbitrar point on the circle. Then, b definition, the distance between (h, k) and (, ) must be r. h k r r (h, k) (, ) h k r Square both sides. Figure  Standard Equation of a Circle The standard equation of a circle, centered at (h, k) with radius r, is given b h k r where r 7 0 Note: If a circle is centered at the origin (0, 0), then h 0 and k 0, and the equation simplifies to r. Eample Graphing a Circle Find the center and radius of each circle. Then graph the circle. a. b. a 0 b 9 c. 0 Solution: a. h, k, and r The center is, and the radius is (Figure ) (, ) r Figure 
4 788 Chapter Conic Sections Calculator Connections Graphing calculators are designed to graph functions, in which is written in terms of. A circle is not a function. However, it can be graphed as the union of two functions one representing the top semicircle and the other representing the bottom semicircle. Solving for in Eample (a), we have Graph these functions as Y and Y, using a square viewing window. Y Y Notice that the image from the calculator does not show the upper and lower semicircles connecting at their endpoints, when in fact the semicircles should hook up. This is due to the calculator s limited resolution. b. c. a 0 b 9 0 a 0 b a b The center is 0, 0 and the radius is (Figure ) The center is (0, 0) and the radius is 0. (Figure ). 0 (0, ) r Figure  (0, 0) r 0 Figure 
5 Section. Distance Formula and Circles 789 Skill Practice Find the center and radius of each circle. Then graph the circle... a 7 b 9 9. Sometimes it is necessar to complete the square to write an equation of a circle in standard form. Eample Writing the Equation of a Circle in the Form ( h) ( k ) r Identif the center and radius of the circle given b the equation 0. Solution: 0 The center is, 8 and the radius is. Skill Practice 8 8 To identif the center and radius, write the equation in the form h k r. Group the terms and group the terms. Move the constant to the righthand side. Complete the square on. Add to both sides of the equation. Complete the square on. Add to both sides of the equation. Factor and simplif.. Identif the center and radius of the circle given b the equation Skill Practice Answers. Center; (, ); radius: 7. Center: (, 0); radius:. Center: (0, 0); radius:. Center: (, ); radius:
6 790 Chapter Conic Sections. Writing an Equation of a Circle Eample Writing an Equation of a Circle Write an equation of the circle shown in Figure 7. Figure 7 Solution: The center is (, ); therefore, h and k. From the graph, r. h k r Skill Practice Skill Practice Answers.. Write an equation for a circle whose center is the point (, ) and whose radius is 8. Section. Boost our GRADE at mathzone.com! Stud Skills Eercise. Define the ke terms. Practice Eercises Practice Problems SelfTests NetTutor eprofessors Videos a. Distance formula b. Circle c. Center d. Radius e. Standard equation of a circle Concept : Distance Formula For Eercises, use the distance formula to find the distance between the two points.., 7 and, 9., 0 and,. 0, and, 8., 7 and,. a and a 7. a and a,,,, b 0 b 8 b b 8. (, ) and, 9., and, , and,. (7, ) and (, )., and,., and,
7 Section. Distance Formula and Circles 79., 7 and, 7., and,., 0 and 0, 7. Eplain how to find the distance between and 7 on the ais. 8. Eplain how to find the distance between and 7 on the ais. 9. Find the values of such that the distance between the points (, 7) and, is 0 units. 0. Find the values of such that the distance between the points, and, is units.. Find the values of such that the distance between the points (, ) and, is units.. Find the values of such that the distance between the points, and (, ) is 0 units. For Eercises, determine if the three points define the vertices of a right triangle..,,,, and (, ).,,,, and (7, ).,,,, and (, ). (, ), (, ), and (, 0) Concept : Circles For Eercises 7 8, identif the center and radius of the circle and then graph the circle. Complete the square, if necessar
8 79 Chapter Conic Sections a a b 9 b
9 Section. Distance Formula and Circles Concept : Writing an Equation of a Circle For Eercises 9, write an equation that represents the graph of the circle Write an equation of a circle centered at the origin with a radius of 7 m.. Write an equation of a circle centered at the origin with a radius of m.
10 79 Chapter Conic Sections 7. Write an equation of a circle centered at, with a diameter of ft. 8. Write an equation of a circle centered at, with a diameter of 8 ft. Epanding Your Skills 9. Write an equation of a circle that has the points (, ) and (, ) as endpoints of a diameter. 0. Write an equation of a circle that has the points (, ) and (, ) as endpoints of a diameter.. Write an equation of a circle whose center. Write an equation of a circle whose is at (, ) and is tangent to the  and center is at, and is tangent to the aes. (Hint: Sketch the circle first.)  and aes. (Hint: Sketch the circle first.) Write an equation of a circle whose center is at (, ) and that passes through the point,.. Write an equation of a circle whose center is at, and that passes through the point,. Graphing Calculator Eercises For Eercises 70, graph the circles from the indicated eercise on a square viewing window, and approimate the center and the radius from the graph.. 9 (Eercise 7). (Eercise 8) 7. (Eercise ) 8. (Eercise ) 9. (Eercise ) 70. (Eercise )
11 Section. More on the Parabola 79 More on the Parabola. Introduction to Conic Sections Recall that the graph of a seconddegree equation of the form a b c (a 0) is a parabola. In Section. we learned that the graph of ( h) ( k) r is a circle. These and two other tpes of figures called ellipses and hperbolas are called conic sections. Conic sections derive their names because each is the intersection of a plane and a doublenapped cone (Figure 8). Section. Concepts. Introduction to Conic Sections. Parabola Vertical Ais of Smmetr. Parabola Horizontal Ais of Smmetr. Verte Formula Circle Parabola Ellipse Hperbola Figure 8. Parabola Vertical Ais of Smmetr A parabola is defined b a set of points in a plane that are equidistant from a fied line (called the directri) and a fied point (called the focus) not on the directri. Parabolas have numerous realworld applications. For eample, a reflecting telescope has a mirror with the cross section in the shape of a parabola. A parabolic mirror has the propert that incoming ras of light are reflected from the surface of the mirror to the focus. Focus Mirror 0 The graph of the solutions to the quadratic equation a b c is a parabola. We graphed parabolas of this tpe in Section 8. b writing the equation in the form a h k. Recall that the verte (h, k) is the highest or
12 79 Chapter Conic Sections the lowest point of a parabola. The ais of smmetr of the parabola is a line that passes through the verte and is perpendicular to the directri (Figure 9). Verte Ais of smmetr Figure 9 Standard Form of the Equation of a Parabola Vertical Ais of Smmetr The standard form of the equation of a parabola with verte h, k and vertical ais of smmetr is a h k where a 0 If a 7 0, then the parabola opens upward; and if a 0, the parabola opens downward. The ais of smmetr is given b h. In Section 8., we learned that it is sometimes necessar to complete the square to write the equation of a parabola in standard form. Eample Graphing a Parabola b First Completing the Square Given: the equation of the parabola a. Write the equation in standard form a h k. b. Identif the verte and ais of smmetr. Determine if the parabola opens upward or downward. c. Graph the parabola. Solution: a. Complete the square to write the equation in the form a h k. Factor out from the variable terms. Add and subtract the quantit. Remove the term from within the parentheses b first appling the distributive propert. When is removed from the parentheses, it carries with it the factor of from outside the parentheses.
13 Section. More on the Parabola 797 The equation is in the form a h k where a, h, and k. b. The verte is,. Because a is negative, a 0, the parabola opens downward. The ais of smmetr is. c. To graph the parabola, we know that its orientation is downward. Furthermore, we know the verte is (, ). To find other points on the parabola, select several values of and solve for. Recall that the intercept is found b substituting 0 and solving for. 0 Choose. Solve for. Verte intercept ( ) (, ) (0, ) (, ) (, ) (, ) Skill Practice. Given: the equation of the parabola a. Write the equation in standard form. b. Identif the verte and ais of smmetr. c. Graph the parabola.. Parabola Horizontal Ais of Smmetr We have seen that the graph of a parabola a b c opens upward if a 7 0 and downward if a 0. A parabola can also open to the left or right. In such a case, the roles of and are essentiall interchanged in the equation. Thus, the graph of a b c opens to the right if a 7 0 (Figure 0) and to the left if a 0 (Figure ). Ais of smmetr Ais of smmetr Verte Verte a b c, a 7 0 a b c, a 0 Figure 0 Figure  Skill Practice Answers a. b. Verte: (, ); ais of smmetr: c.
14 798 Chapter Conic Sections Standard Form of the Equation of a Parabola Horizontal Ais of Smmetr The standard form of the equation of a parabola with verte h, k and horizontal ais of smmetr is a k h where a 0 If a 7 0, then the parabola opens to the right and if a 0, the parabola opens to the left. The ais of smmetr is given b k. Eample Graphing a Parabola with a Horizontal Ais of Smmetr Given the equation of the parabola, a. Determine the coordinates of the verte and the equation of the ais of smmetr. b. Use the value of a to determine if the parabola opens to the right or left. c. Plot several points and graph the parabola. Solution: a. The equation can be written in the form a k h: 0 0 Therefore, h 0 and k 0. The verte is (0, 0). The ais of smmetr is 0 (the ais). b. Because a is positive a 7 0, the parabola opens to the right. c. The verte of the parabola is (0, 0). To find other points on the graph, select values for and solve for. Skill Practice Answers a. Verte: (, 0); ais of smmetr: 0 b. Parabola opens left. c. Skill Practice Solve for. 0 0 Choose. Verte (0, 0) (, ) (, ). Given: the equation a. Identif the verte and the ais of smmetr. b. Determine if the parabola opens to the right or to the left. c. Graph the parabola. 7 8
15 Section. More on the Parabola 799 Calculator Connections Graphing calculators are designed to graph functions, where is written as a function of. The parabola from Eample is not a function. It can be graphed, however, as the union of two functions, one representing the top branch and the other representing the bottom branch. Solving for in Eample, Graph these functions as Y we have: and Y. B Y and Y Eample Graphing a Parabola b First Completing the Square Given the equation of the parabola 8, a. Write the equation in standard form = a( k) + h. b. Identif the verte and ais of smmetr. Determine if the parabola opens to the right or left. c. Graph the parabola. Solution: a. Complete the square to write the equation in the form a k h Factor out from the variable terms. Add and subtract the quantit 8. Remove the term from within the parentheses b first appling the distributive propert. When is removed from the parentheses, it carries with it the factor of from outside the parentheses. The equation is in the form a k h, where a, h, and k.
16 800 Chapter Conic Sections b. The verte is,. Because a is negative a 0, the parabola opens to the left. The ais of smmetr is. c. Solve for. Choose. Verte (, ) 7 (, ) (, ) (, ) (, ) ( ) Skill Practice. Given the equation of the parabola 8, a. Determine the coordinates of the verte. b. Determine if the parabola opens to the right or to the left. c. Graph the parabola.. Verte Formula From Section 8., we learned that the verte formula can also be used to find the verte of a parabola. For a parabola defined b a b c, The coordinate of the verte is given b b a. The coordinate of the verte is found b substituting this value for into the original equation and solving for. For a parabola defined b a b c, The coordinate of the verte is given b b a. The coordinate of the verte is found b substituting this value for into the original equation and solving for. Skill Practice Answers a. Verte: (, ) b. Right c Eample Finding the Verte of a Parabola b Using the Verte Formula Find the verte b using the verte formula. a. b. Solution: a. The parabola is in the form a b c. a b c Identif a, b, and c.
17 Section. More on the Parabola 80 The coordinate of the verte is given b The coordinate of the verte is found b substitution:. The verte is,. b. The parabola is in the form a b c. a b c Identif a, b, and c. The coordinate of the verte is given b b a. b a. The coordinate of the verte is found b substitution. 9 9 The verte is, Skill Practice Find the verte b using the verte formula... Skill Practice Answers. Verte: a9, b. Verte:, Section. Boost our GRADE at mathzone.com! Stud Skills Eercise. Define the ke terms. Practice Eercises Practice Problems SelfTests NetTutor eprofessors Videos a. Conic sections b. Parabola c. Ais of smmetr d. Verte Review Eercises. Determine the distance between the points (, ) and,.. Determine the distance from the origin to the point,.. Determine the distance between the points, and,.
18 80 Chapter Conic Sections For Eercises 8, identif the center and radius of the circle and then graph the circle Concept : Parabola Vertical Ais of Smmetr For Eercises 9, use the equation of the parabola in standard form a h k to determine the coordinates of the verte and the equation of the ais of smmetr. Then graph the parabola
19 Section. More on the Parabola Concept : Parabola Horizontal Ais of Smmetr For Eercises 7, use the equation of the parabola in standard form a k h to determine the coordinates of the verte and the ais of smmetr. Then graph the parabola
20 80 Chapter Conic Sections Concept : Verte Formula For Eercises, determine the verte b using the verte formula Mied Eercises. Eplain how to determine whether a parabola opens upward, downward, left, or right.. Eplain how to determine whether a parabola has a vertical or horizontal ais of smmetr. For Eercises 7 8, use the equation of the parabola first to determine whether the ais of smmetr is vertical or horizontal. Then determine if the parabola opens upward, downward, left, or right. 7. ( ) 8. ( ) 9. ( ) 0. ( )... ( ). ( ) Section. Concepts. Standard Form of an Equation of an Ellipse. Standard Forms of an Equation of a Hperbola The Ellipse and Hperbola. Standard Form of an Equation of an Ellipse In this section we will stud the two remaining conic sections: the ellipse and the hperbola. An ellipse is the set of all points (, ) such that the sum of the distance between (, ) and two distinct points is a constant. The fied points are called the foci (plural of focus) of the ellipse. To visualize an ellipse, consider the following application. Suppose Sona wants to cut an elliptical rug from a rectangular rug to avoid a stain made b the famil dog. She places two tacks along the center horizontal line. Then she ties the ends of a slack piece of rope to each tack.with the rope pulled tight, she traces out a curve.this curve is an ellipse, and the tacks are located at the foci of the ellipse (Figure ). Stain Tack Tack Figure  We will first graph ellipses that are centered at the origin.
21 Section. The Ellipse and Hperbola 80 Standard Form of an Equation of an Ellipse Centered at the Origin An ellipse with the center at the origin has the equation where a b a and b are positive real numbers. In the standard form of the equation, the right side must equal. To graph an ellipse centered at the origin, find the  and intercepts. To find the intercepts, let 0. To find the intercepts, let 0. a 0 b 0 a b a b a b (0, b) a a The intercepts are (a, 0) and a, 0. b b The intercepts are (0, b) and 0, b. ( a, 0) (0, 0) (a, 0) (0, b) Eample Graphing an Ellipse Graph the ellipse given b the equation. 9 Solution: The equation can be written as therefore, a and b. ; Graph the intercepts (, 0),, 0, (0, ), 0, and sketch the ellipse. Skill Practice 9 (0, ) (, 0) (, 0) (0, ). Graph the ellipse given b the equation. Skill Practice Answers.
22 80 Chapter Conic Sections Eample Graphing an Ellipse Graph the ellipse given b the equation. Solution: First, to write the equation in standard form, divide both sides b : The equation can then be written as therefore, a and b. ; Graph the intercepts (, 0),, 0, (0, ), and 0, and sketch the ellipse. (, 0) (0, ) (, 0) (0, ) Skill Practice. Graph the ellipse given b the equation 00. Skill Practice Answers A circle is a special case of an ellipse where a b. Therefore, it is not surprising that we graph an ellipse centered at the point (h, k) in much the same wa we graph a circle. Eample Graph the ellipse give b the equation. Graphing an Ellipse Whose Center is Not at the Origin Solution: Just as we would find the center of a circle, we see that the center of the ellipse is,. Now we can use the values of a and b to help us plot four strategic points to define the curve. The equation can be written as. From this, we have a and b. To sketch the curve, locate the center at,. Then move a 7 (, ) units to the left and right of the center and plot a two points. Similarl, move b units up and b down from the center and plot two points. Skill Practice. Graph the ellipse. 9 7
23 Section. The Ellipse and Hperbola 807. Standard Forms of an Equation of a Hperbola A hperbola is the set of all points (, ) such that the difference of the distances between (, ) and two distinct points is a constant. The fied points are called the foci of the hperbola. The graph of a hperbola has two parts, called branches. Each part resembles a parabola but is a slightl different shape. A hperbola has two vertices that lie on an ais of smmetr called the transverse ais. For the hperbolas studied here, the transverse ais is either horizontal or vertical. Transverse ais Transverse ais Vertices Vertices Standard Forms of an Equation of a Hperbola with Center at the Origin Let a and b represent positive real numbers. Horizontal transverse ais: The standard form of an equation of a hperbola with a horizontal transverse ais and center at the origin is given b a b. Note: The term is positive. The branches of the hperbola open left and right. Vertical transverse ais: The standard form of an equation of a hperbola with a vertical transverse ais and center at the origin is given b b a. Note: The term is positive. The branches of the hperbola open up and down. In the standard forms of an equation of a hperbola, the right side must equal. To graph a hperbola centered at the origin, first construct a reference rectangle. Draw this rectangle b using the points (a, b), a, b, a, b, and a, b. Asmptotes lie on the diagonals of the rectangle. The branches of the hperbola are drawn to approach the asmptotes. ( a, b) ( a, b) (a, b) (a, b) ( a, b) (a, b) ( a, b) (a, b)
24 808 Chapter Conic Sections Eample Graphing a Hperbola Graph the hperbola given b the equation. 9 a. Determine whether the transverse ais is horizontal or vertical. b. Draw the reference rectangle and asmptotes. c. Graph the hperbola and label the vertices. TIP: In the equation, the 9 term is positive. Therefore, the hperbola opens in the direction (left/right). Solution: a. Since the term is positive, the transverse ais is horizontal. b. The equation can be written therefore, a and b. Graph ; the reference rectangle from the points (, ),,,,,, Asmptote Verte Verte (, 0) (, 0) Asmptote c. The coordinates of the vertices are, 0 and (, 0). Skill Practice. Graph the hperbola 9. Eample Graphing a Hperbola Graph the hperbola given b the equation. 0 a. Write the equation in standard form to determine whether the transverse ais is horizontal or vertical. b. Draw the reference rectangle and asmptotes. c. Graph the hperbola and label the vertices. Solution: Skill Practice Answers. a. Isolate the variable terms and divide b :. Since the term is positive, the transverse ais is vertical. b. The equation can be written ; therefore, a and b. Graph the reference rectangle from the points (, ),,,,,,. Verte (0, ) Asmptote Verte Asmptote (0, )
25 Section. The Ellipse and Hperbola 809 c. The coordinates of the vertices are (0, ) and Skill Practice. Graph the hperbola 9. 0,. Skill Practice Answers. Section. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems SelfTests NetTutor eprofessors Videos. Define the ke terms. a. Ellipse b. Hperbola c. Transverse ais of a hperbola Review Eercises. Write the standard form of a circle with center at (h, k) and radius, r. For Eercises, identif the center and radius of the circle For Eercises, identif the verte and the ais of smmetr Write an equation for a circle whose center has coordinates, with radius equal to. 8. Write the equation for the circle centered at the origin and with radius equal to 8. Concept : Standard Form of an Equation of an Ellipse For Eercises 9, find the  and intercepts and graph the ellipse
26 80 Chapter Conic Sections For Eercises 7, identif the center (h, k) of the ellipse. Then graph the ellipse
27 Section. The Ellipse and Hperbola Concept : Standard Forms of an Equation of a Hperbola For Eercises, determine whether the transverse ais is horizontal or vertical For Eercises 0, use the equation in standard form to graph the hperbola. Label the vertices of the hperbola
28 8 Chapter Conic Sections Mied Eercises For Eercises, determine if the equation represents an ellipse or a hperbola Epanding Your Skills. An arch for a tunnel is in the shape of a semiellipse. The distance between vertices is 0 ft, and the height to the top of the arch is 0 ft. Find the height of the arch 0 ft from the center. Round to the nearest foot. 0 ft 0 ft. A bridge over a gorge is supported b an arch in the shape of a semiellipse. The length of the bridge is 00 ft, and the height is 00 ft. Find the height of the arch 0 ft from the center. Round to the nearest foot. 00 ft 00 ft
29 Section. Nonlinear Sstems of Equations in Two Variables 8 In Eercises 8, graph the hperbola centered at (h, k) b following these guidelines. First locate the center. Then draw the reference rectangle relative to the center (h, k). Using the reference rectangle as a guide, sketch the hperbola. Label the center and vertices Nonlinear Sstems of Equations in Two Variables. Solving Nonlinear Sstems of Equations b the Substitution Method Recall that a linear equation in two variables and is an equation that can be written in the form a b c, where a and b are not both zero. In Sections.., we solved sstems of linear equations in two variables b using the graphing method, the substitution method, and the addition method. In this section, we will solve nonlinear sstems of equations b using the same methods. A nonlinear sstem of equations is a sstem in which at least one of the equations is nonlinear. Graphing the equations in a nonlinear sstem helps to determine the number of solutions and to approimate the coordinates of the solutions. The substitution method is used most often to solve a nonlinear sstem of equations analticall. Section. Concepts. Solving Nonlinear Sstems of Equations b the Substitution Method. Solving Nonlinear Sstems of Equations b the Addition Method
30 8 Chapter Conic Sections Eample Solving a Nonlinear Sstem of Equations Given the sstem a. Solve the sstem b graphing. b. Solve the sstem b the substitution method. Solution: a. 7 is a line (the slopeintercept form is is a circle centered at the origin with radius. From Figure , we appear to have two solutions, and (, ). 7 (, ) (, ) Figure  b. To use the substitution method, isolate one of the variables from one of the equations. We will solve for in the first equation. A B Solve for. 7 7 B Substitute 7 for in the second equation. The resulting equation is quadratic in. Set the equation equal to zero. Factor. For each value of, find the corresponding value from the equation 7. : : or 7 7 The solution is,. The solution is (, ). (See Figure .)
31 Section. Nonlinear Sstems of Equations in Two Variables 8 Skill Practice. Given the sstem 0 a. Solve the sstem b graphing. b. Solve the sstem b the substitution method. Eample Given the sstem a. Sketch the graphs. Solving a Nonlinear Sstem b the Substitution Method b. Solve the sstem b the substitution method. Solution: a. is one of the si basic functions graphed in Section is a circle centered at the origin with radius 0.. From Figure , we see that there is one solution. Figure  + = 0 = b. To use the substitution method, we will substitute into equation B. A B 0 B Substitute into the second equation. Set the second equation equal to zero. Factor. or Reject because it is not in the domain of. Substitute into the equation. If, then. The solution is (, ). Skill Practice Answers a b., 7 and, 8 0
32 8 Chapter Conic Sections Skill Practice. Given the sstem 8 a. Sketch the graphs. b. Solve the sstem b using the substitution method. Calculator Connections Graph the equations from Eample to confirm our solution to the sstem of equations. Use an Intersect feature or Zoom and Trace to approimate the point of intersection. Recall that the circle must be entered into the calculator as two functions: Y 0 Y 0 Y Calculator Connections Graph the equations and to support the solutions to Eample. Skill Practice Answers a. b., (, ) (0, 0) (, ) Y Y Eample Solving a Nonlinear Sstem b Using the Substitution Method Solve the sstem b using the substitution method. Solution: A B Because is isolated in both equations, we can equate the epressions for. To solve the radical equation, raise both sides to the third power. This is a thirddegree polnomial equation. 0 Set the equation equal to zero. 0 Factor out the GCF. 0 Factor completel. 0 or or For each value of, find the corresponding value from either original equation. We will use equation B :. If 0, then 0. The solution is (0, 0). If, then. The solution is,. If, then. T
33 Section. Nonlinear Sstems of Equations in Two Variables 87 Skill Practice. Solve the sstem b using the substitution method. 9. Solving Nonlinear Sstems of Equations b the Addition Method The substitution method is used most often to solve a sstem of nonlinear equations. In some situations, however, the addition method offers an efficient means of finding a solution. Eample demonstrates that we can eliminate a variable from both equations provided the terms containing that variable are like terms. Eample Solve the sstem. Solving a Nonlinear Sstem of Equations b the Addition Method 7 Solution: A B A B Notice that the To eliminate the b. Multipl b. terms are like in each equation. terms, multipl the first equation Eliminate the term. Substitute each value of into one of the original equations to solve for. We will use equation A : 7. : : 7 7 A A The solutions are (, ) and,. The solutions are, and,. TIP: In Eample, the terms are also like in both equations. We could have eliminated the  terms b multipling equation B b. Skill Practice Answers. 0, 0;, ;,
34 88 Chapter Conic Sections Skill Practice. Solve the sstem b using the addition method. 7 Calculator Connections The solutions to Eample can be checked from the graphs of the equations. For the equation 7, we have 7 For the equation, we have B (, ) (, ) (, ) (, ) TIP: It is important to note that the addition method can be used onl if two equations share a pair of like terms. The substitution method is effective in solving a wider range of sstems of equations. The sstem in Eample could also have been solved b using substitution. A 7 Solve for. 7 B B 7 : 7 9 : 7 9 The solutions are (, ) and,. The solutions are, and,. Skill Practice Answers., ;, ;, ;,
35 Section. Nonlinear Sstems of Equations in Two Variables 89 Section. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems SelfTests NetTutor. Define the ke term nonlinear sstem of equations. eprofessors Videos Review Eercises. Write the distance formula between two points (, ) and (, ) from memor.. Find the distance between the two points 8, and, 8.. Write an equation representing the set of all points units from the point. Write an equation representing the set of all points 8 units from the point,.,. For Eercises, determine if the equation represents a parabola, circle, ellipse, or hperbola Concept : Solving Nonlinear Sstems of Equations b the Substitution Method For Eercises, use sketches to eplain.. How man points of intersection are possible between a line and a parabola?. How man points of intersection are possible between a line and a circle?. How man points of intersection are possible between two distinct circles? 7. How man points of intersection are possible between two distinct parabolas of the form a b c, a 0? 8. How man points of intersection are possible between a circle and a parabola? 9. How man points of intersection are possible between two distinct lines? 0. How man points of intersection are possible with an ellipse and a hperbola?. How man points of intersection are possible with an ellipse and a parabola?
36 80 Chapter Conic Sections For Eercises 7, sketch each sstem of equations. Then solve the sstem b the substitution method For Eercises 8, solve the sstem b the substitution method Concept : Solving Nonlinear Sstems of Equations b the Addition Method For Eercises 7 0, solve the sstem of nonlinear equations b the addition method
37 Section. Nonlinear Sstems of Equations in Two Variables Epanding Your Skills. The sum of two numbers is 7. The sum of the squares of the numbers is. Find the numbers.. The sum of the squares of two numbers is 00. The sum of the numbers is. Find the numbers.. The sum of the squares of two numbers is. The difference of the squares of the numbers is 8. Find the numbers.. The sum of the squares of two numbers is. The difference of the squares of the numbers is 8. Find the numbers. Graphing Calculator Eercises For Eercises 8, use the Intersect feature or Zoom and Trace to approimate the solutions to the sstem.. (Eercise ). (Eercise ) 9 7. (Eercise 0) 8. (Eercise ) For Eercises 9 0, graph the sstem on a square viewing window. What can be said about the solution to the sstem? 9. 0.
38 8 Chapter Conic Sections Section. Concepts. Nonlinear Inequalities in Two Variables. Sstems of Nonlinear Inequalities in Two Variables Nonlinear Inequalities and Sstems of Inequalities. Nonlinear Inequalities in Two Variables In Section 9. we graphed the solution sets to linear inequalities in two variables, such as. This involved graphing the related equation (a line in the plane) and then shading the appropriate region above or below the line. See Figure . In this section, we will emplo the same technique to solve nonlinear inequalities in two variables. Figure  Eample Graphing a Nonlinear Inequalit in Two Variables Graph the solution set of the inequalit. Solution: The related equation is a circle of radius, centered at the origin. Graph the related equation b using a dashed curve because the points satisfing the equation are not part of the solution to the strict inequalit. See Figure . Figure  Notice that the dashed curve separates the plane into two regions, one inside the circle, the other outside the circle. Select a test point from each region and test the point in the original inequalit. Test Point Inside : (0, 0) Test Point Outside : (, ) 0 0? 0? True?? False
39 Section. Nonlinear Inequalities and Sstems of Inequalities 8 The inequalit is true at the test point (0, 0). Therefore, the solution set is the region inside the circle. See Figure 7. Figure 7 Skill Practice. Graph the solution set of the inequalit 9. Eample Graphing a Nonlinear Inequalit in Two Variables Graph the solution set of the inequalit 9. Solution: First graph the related equation 9. Notice that the equation can be written in the standard form of a hperbola. 9 9 Subtract from both sides. 9 Divide both sides b. Graph the hperbola as a solid curve, because the original inequalit includes equalit. See Figure 8. Figure 8 9 The hperbola divides the plane into three regions: a region above the upper branch, a region between the branches, and a region below the lower branch. Select a test point from each region. 9 Skill Practice Answers.
40 8 Chapter Conic Sections Test: (0, ) Test: (0, 0) Test: (0, ) 9? 0 90? 0 9? 0 8? True 0? False 8? True Shade the regions above the top branch and below the bottom branch of the hperbola. See Figure Figure 9 Skill Practice. Graph the solution set of the inequalit 9. Skill Practice Answers.. Sstems of Nonlinear Inequalities in Two Variables In Section. we solved sstems of nonlinear equations in two variables. The solution set for such a sstem is the set of ordered pairs that satisf both equations simultaneousl. We will now solve sstems of nonlinear inequalities in two variables. Similarl, the solution set is the set of all ordered pairs that simultaneousl satisf each inequalit. To solve a sstem of inequalities, we will graph the solution to each individual inequalit and then take the intersection of the solution sets. Eample Graph the solution set. Graphing a Sstem of Nonlinear Inequalities in Two Variables 7 e Solution: The solution to 7 e is the set of points above the curve e. See Figure 0. The solution to is the set of points below the parabola. See Figure . Figure 0 e Figure 
41 Section. Nonlinear Inequalities and Sstems of Inequalities 8 The solution to the sstem of inequalities is the intersection of the solution sets of the individual inequalities. See Figure . Skill Practice. Graph the solution set. e Figure Skill Practice Answers. Section. Boost our GRADE at mathzone.com! Review Eercises Practice Eercises Practice Problems SelfTests NetTutor For Eercises, match the equation with its graph. eprofessors Videos. a... b e log 0.. a. b. c. d.
42 8 Chapter Conic Sections e. f. g. h. i. j. k. Concept : Nonlinear Inequalities in Two Variables. True or false? The point (, ) satisfies the inequalit 7.. True or false? The point (, ) satisfies the inequalit.. True or false? The point (, ) satisfies the sstem of inequalities.. True or false? The point (, ) satisfies the sstem of inequalities.. True or false? The point (, ) satisfies the sstem of inequalities. 7. a. Graph the solution set of. 8. a. Graph the solution set of
43 Section. Nonlinear Inequalities and Sstems of Inequalities 87 b. Describe the solution set for the inequalit b. Describe the solution set for the inequalit c. Describe the solution set of the equation c. Describe the solution set for the equation a. Graph the solution set of 0. a. Graph the solution set of.. 9 b. How would the solution change for the b. How would the solution change for the strict inequalit 7? strict inequalit? 9 For Eercises, graph the solution set
44 88 Chapter Conic Sections ln. log.. a 7 b
45 Section. Nonlinear Inequalities and Sstems of Inequalities 89 Concept : Sstems of Nonlinear Inequalities in Two Variables For Eercises 7 0, graph the solution set to the sstem of inequalities
46 80 Chapter Conic Sections e Epanding Your Skills For Eercises, graph the compound inequalities.. or 0. or. or. or
47 Summar 8 Chapter SUMMARY Section. Distance Formula and Circles Ke Concepts The distance between two points, and, is d Eamples Eample Find the distance between two points., and, d The standard equation of a circle with center (h, k) and radius r is h k r Eample Find the center and radius of the circle The center is, and the radius is. Section. More on the Parabola Ke Concepts A parabola is the set of points in a plane that are equidistant from a fied line (called the directri) and a fied point (called the focus) not on the directri. The standard form of an equation of a parabola with verte (h, k) and vertical ais of smmetr is a( h) k The equation of the ais of smmetr is h. If a 0, then the parabola opens upward. If a 0, the parabola opens downward. Eamples Eample Given the parabola, The verte is (, ). The ais of smmetr is. Verte (, ) =
48 8 Chapter Conic Sections The standard form of an equation of a parabola with verte (h, k) and horizontal ais of smmetr is a( k) h The equation of the ais of smmetr is k. If a 0, then the parabola opens to the right. If a 0, the parabola opens to the left. Eample Given the parabola, determine the coordinates of the verte and the equation of the ais of smmetr. 0 The verte is (, 0). The ais of smmetr is 0. (, 0) 0 Section. The Ellipse and Hperbola Ke Concepts An ellipse is the set of all points (, ) such that the sum of the distances between (, ) and two distinct points (called foci) is constant. Eamples Eample Standard Form of an Ellipse with Center at the Origin An ellipse with the center at the origin has the equation a b where a and b are positive real numbers. ( a, 0) (0, b) (0, 0) (a, 0) (0, b) For an ellipse centered at the origin, the intercepts are given b (a, 0) and ( a, 0), and the intercepts are given b (0, b) and (0, b). A hperbola is the set of all points (, ) such that the difference of the distances between (, ) and two distinct points is a constant. The fied points are called the foci of the hperbola. Eample 9 (0, ) (, 0) (, 0) (0, )
49 Summar 8 Standard Forms of an Equation of a Hperbola Let a and b represent positive real numbers. Horizontal Transverse Ais. The standard form of an equation of a hperbola with a horizontal transverse ais and center at the origin is given b a b Eample (, 0) (, 0) Vertical Transverse Ais. The standard form of an equation of a hperbola with a vertical transverse ais and center at the origin is given b a b Eample (0, ) (0, ) Section. Ke Concepts A nonlinear sstem of equations can be solved b graphing or b the substitution method. 0 (, ) (, ) Nonlinear Sstems of Equations in Two Variables Eamples Eample A B A Solve for : Substitute in first equation. 0 or 0 or If then. If then. Points of intersection are, and,.
50 8 Chapter Conic Sections A nonlinear sstem ma also be solved b using the addition method when the equations share like terms. Eample Mult. b If, If, The points of intersection are,, and,.,,,, Section. Nonlinear Inequalities and Sstems of Inequalities Ke Concepts Graph a nonlinear inequalit b using the test point method.that is, graph the related equation.then choose test points in each region to determine where the inequalit is true. Eamples Eample Eample
51 Review Eercises 8 Graph a sstem of nonlinear inequalities b finding the intersection of the solution sets. That is, graph the solution set for each individual inequalit, then take the intersection. Eample 0, 7, and Chapter Review Eercises Section. For Eercises, find the distance between the two points b using the distance formula.., and (0, ). (, ) and,. Find such that (, ) is units from (, 9).. Find such that, is units from (, ). Points are said to be collinear if the all lie on the same line. If three points are collinear, then the distance between the outermost points will equal the sum of the distances between the middle point and each of the outermost points. For Eercises, determine if the three points are collinear..,,,, and (, ). A stained glass window is in the shape of a circle with a in. diameter. Find an equation of the circle relative to the origin for each of the following graphs. a. b. in. For Eercises, write the equation of the circle in standard form b completing the square. in..,, 0,, and, For Eercises 7 0, find the center and the radius of the circle Write an equation of a circle with center at the origin and a diameter of 7 m. 7. Write an equation of a circle with center at (0, ) and a diameter of m.
52 8 Chapter Conic Sections Section. For Eercises 8, determine whether the ais of smmetr is vertical or horizontal and if the parabola opens upward, downward, left, or right. 8. ( ) 9. ( 9) 0. ( ) 8. ( ) 0 For Eercises, determine the coordinates of the verte and the equation of the ais of smmetr. Then use this information to graph the parabola... ( ).. For Eercises 9, write the equation in standard form a( h) k or a( k) h. Then identif the verte, and ais of smmetr Section. For Eercises 0, identif the  and Then graph the ellipse intercepts. For Eercises, identif the center of the ellipse and graph the ellipse
53 Review Eercises 87 For Eercises 7, determine whether the transverse ais is horizontal or vertical For Eercises 8 9, graph the hperbola b first drawing the reference rectangle and the asmptotes. Label the vertices For Eercises 0, identif the equations as representing an ellipse or a hperbola Section. For Eercises 7, a. Identif each equation as representing a line, a parabola, a circle, an ellipse, or a hperbola. b. Graph both equations on the same coordinate sstem. c. Solve the sstem analticall and verif the answers from the graph
54 88 Chapter Conic Sections For Eercises 8, solve the sstem of nonlinear equations b using either the substitution method or the addition method Section. 8 For Eercises, graph the solution set to the inequalit For Eercises, graph the solution set to the sstem of nonlinear inequalities
55 Test 89 Chapter Test. Determine the coordinates of the verte and the equation of the ais of smmetr. Then graph the parabola.. Determine the coordinates of the center and radius of the circle. a b a b 9. Use the distance formula to find the distance between the two points, 9 and,.. Determine the center and radius of the circle.. Write the equation in standard form a( h) k, and graph the parabola. 0. Let (0, ) be the center of a circle that passes through the point,. a. What is the radius of the circle? b. Write the equation of the circle in standard form. 7. Graph the ellipse Graph the ellipse. 9. Graph the hperbola
56 80 Chapter Conic Sections 0. Solve the sstems and identif the correct graph of the equations. a. 9 b. i. ii.. Describe the circumstances in which a nonlinear sstem of equations can be solved b using the addition method.. Solve the sstem b using either the substitution method or the addition method For Eercises, graph the solution set Chapters Cumulative Review Eercises. Solve the equation.. Solve the inequalit. Graph the solution and write the solution in interval notation.. The product of two integers is 0. If one integer is less than twice the other, find the integers.. For 0: a. Find the  and intercepts. b. Find the slope. c. Graph the line The amount of mone spent on motor vehicles and parts each ear since the ear 000 is shown in the graph. Let = 0 correspond to the 7 ear 000. Let represent the amount of mone spent on motor vehicles and parts (in billions of dollars). a. Use an two data points to find the slope of the line. b. Find an equation of the line through the points. Write the answer in slopeintercept form. c. Use the linear model found in part (b) to predict the amount spent on motor vehicles and parts in the ear 00. Dollars (in billions) Amount Spent (in $ billions) on Motor Vehicles and Parts in United States (0, 90) (, 0) (, 0) (, 0) Year ( 0 corresponds to 000) Source: Bureau of Economic Analsis, U.S. Department of Commerce
57 Cumulative Review Eercises 8. Find the slope and intercept of b first writing the equation in slopeintercept form. 7. A collection of dimes and quarters has a total value of $.. If there are 7 coins, how man of each tpe are there? 8. Solve the sstem. z 9. a. Given the matri A c, 0 ` 8 d write the matri obtained b multipling the elements in the second row b. b. Using the matri obtained from part (a), write the matri obtained b multipling the second row b and adding the result to the first row. 0. Solve the following sstem Solve using Cramer s rule: z. For f, find the function values f 0, f, f, and f().. For g,, 8,,, 0,, find the function values g(), g(8), g(), and g.. The quantit z varies jointl as and as the square of. If z is 80 when is and is, find z when and.. For f and g, find g f.. a. Find the value of the epression for. b. Factor the epression and find the value when is. c. Compare the values for parts (a) and (b). 7. Factor completel. 8. Multipl: 9. Solve for : Reduce the epression:. Subtract:. Solve:. Solve the radical equations. a. b.. Perform the indicated operations with comple numbers. a. i i b.. Find the length of the missing side. 0 m a a a 8a? m. An automobile starts from rest and accelerates at a constant rate for 0 sec. The distance, d(t), in feet traveled b the car is given b d(t) =.t i where 0 t 0 is the time in seconds. a. How far has the car traveled after,, and sec, respectivel? b. How long will it take for the car to travel 8. ft? 7. Solve the equation w 0 b factoring and using the quadratic formula. (Hint: You will find one real solution and two imaginar solutions.)
58 8 Chapter Conic Sections 8. Solve the rational equation. 9. Find the coordinates of the verte of the parabola defined b f 0 b completing the square. 0. Graph the quadratic function defined b g. a. Label the intercepts. b. Label the intercept. c. Label the verte.. Solve the inequalit and write the answer in interval notation Solve the inequalit: 0 0. Write the epression in logarithmic form. 8 /. Solve the equation:. For h, find h.. Write an equation representing the set of all points units from the point (0, ). 7. Can a circle and a parabola intersect in onl one point? Eplain. 8. Solve the sstem of nonlinear equations. 9. Graph the solution set. 0. Graph the solution set to this sstem. 7 a b 0
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