# REVIEW OF CONIC SECTIONS

Size: px
Start display at page:

## Transcription

1 REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result from intersecting a cone with a plane as shown in Figure. ellipse parabola hperbola FIGURE Conics PARABOLAS ais focus verte FIGURE F(, p) F parabola directri P(, ) A parabola is the set of points in a plane that are equidistant from a fied point F (called the focus) and a fied line (called the directri). This definition is illustrated b Figure. Notice that the point halfwa between the focus and the directri lies on the parabola; it is called the verte. The line through the focus perpendicular to the directri is called the ais of the parabola. In the 6th centur Galileo showed that the path of a projectile that is shot into the air at an angle to the ground is a parabola. Since then, parabolic shapes have been used in designing automobile headlights, reflecting telescopes, and suspension bridges. (See Challenge Problem.4 for the reflection propert of parabolas that makes them so useful.) We obtain a particularl simple equation for a parabola if we place its verte at the origin O and its directri parallel to the -ais as in Figure. If the focus is the point, p, then the directri has the equation p. If P, is an point on the parabola, then the distance from P to the focus is O =_p p PF s p p and the distance from P to the directri is. (Figure illustrates the case where p.) The defining propert of a parabola is that these distances are equal: FIGURE s p p We get an equivalent equation b squaring and simplifing: p p p p p p p 4p Thomson Brooks-Cole copright 7 An equation of the parabola with focus, p and directri p is 4p

2 REVIEW OF CONIC SECTIONS If we write a 4p, then the standard equation of a parabola () becomes a. It opens upward if p and downward if p [see Figure 4, parts (a) and (b)]. The graph is smmetric with respect to the -ais because () is unchanged when is replaced b. (, p) =_p (, p) =_p =_p ( p, ) (p, ) =_p (a) =4p, p> (b) =4p, p< (c) =4p, p> (d) =4p, p< FIGURE 4 If we interchange and in (), we obtain += 5 _, FIGURE 5 = 5 4p which is an equation of the parabola with focus p, and directri p. (Interchanging and amounts to reflecting about the diagonal line.) The parabola opens to the right if p and to the left if p [see Figure 4, parts (c) and (d)]. In both cases the graph is smmetric with respect to the -ais, which is the ais of the parabola. EXAMPLE Find the focus and directri of the parabola and sketch the graph. SOLUTION If we write the equation as and compare it with Equation, we see that 4p, so p 5. Thus the focus is p, ( 5, ) and the directri is 5. The sketch is shown in Figure 5. ELLIPSES F F FIGURE 6 FIGURE 7 P P(, ) F (_c, ) F (c, ) An ellipse is the set of points in a plane the sum of whose distances from two fied points F and F is a constant (see Figure 6). These two fied points are called the foci (plural of focus). One of Kepler s laws is that the orbits of the planets in the solar sstem are ellipses with the Sun at one focus. In order to obtain the simplest equation for an ellipse, we place the foci on the -ais at the points c, and c, as in Figure 7 so that the origin is halfwa between the foci. Let the sum of the distances from a point on the ellipse to the foci be a. Then P, is a point on the ellipse when PF PF a that is, s c s c a or s c a s c Squaring both sides, we have c c 4a 4as c c c Thomson Brooks-Cole copright 7 which simplifies to as c a c We square again: a c c a 4 a c c which becomes a c a a a c

3 REVIEW OF CONIC SECTIONS (_a, ) (_c, ) FIGURE 8 + = (, b) b a (a, ) c (c, ) (, _b) From triangle F F P in Figure 7 we see that c a, so c a and, therefore, a c. For convenience, let b a c. Then the equation of the ellipse becomes b a a b or, if both sides are divided b a b, a b Since b a c a, it follows that b a. The -intercepts are found b setting. Then a, or a, so a. The corresponding points a, and a, are called the vertices of the ellipse and the line segment joining the vertices is called the major ais. To find the -intercepts we set and obtain b, so b. Equation is unchanged if is replaced b or is replaced b, so the ellipse is smmetric about both aes. Notice that if the foci coincide, then c, so a b and the ellipse becomes a circle with radius r a b. We summarize this discussion as follows (see also Figure 8). (, a) 4 The ellipse a b a b (_b, ) FIGURE 9 + =, a b (, ) (, c) (, _c) (, _a) (b, ) has foci c,, where c a b, and vertices a,. If the foci of an ellipse are located on the -ais at, c, then we can find its equation b interchanging and in (4). (See Figure 9.) 5 The ellipse b a has foci, c, where c a b, and vertices, a. EXAMPLE Sketch the graph of and locate the foci. SOLUTION Divide both sides of the equation b 44: a b (_4, ) {_œ 7, } (4, ) {œ 7, } (, _) 6 9 The equation is now in the standard form for an ellipse, so we have a 6, b 9, a 4, and b. The -intercepts are 4 and the -intercepts are. Also, c a b 7, so c s7 and the foci are (s7, ). The graph is sketched in Figure. FIGURE 9 +6 =44 EXAMPLE Find an equation of the ellipse with foci, and vertices,. SOLUTION Using the notation of (5), we have c and a. Then we obtain b a c 9 4 5, so an equation of the ellipse is 5 9 Thomson Brooks-Cole copright 7 Another wa of writing the equation is Like parabolas, ellipses have an interesting reflection propert that has practical consequences. If a source of light or sound is placed at one focus of a surface with elliptical cross-sections, then all the light or sound is reflected off the surface to the other focus (see

4 4 REVIEW OF CONIC SECTIONS Eercise 59). This principle is used in lithotrips, a treatment for kidne stones. A reflector with elliptical cross-section is placed in such a wa that the kidne stone is at one focus. High-intensit sound waves generated at the other focus are reflected to the stone and destro it without damaging surrounding tissue. The patient is spared the trauma of surger and recovers within a few das. HYPERBOLAS FIGURE P is on the hperbola when PF - PF =a P(, ) F (_c, ) F (c, ) A hperbola is the set of all points in a plane the difference of whose distances from two fied points F and F (the foci) is a constant. This definition is illustrated in Figure. Hperbolas occur frequentl as graphs of equations in chemistr, phsics, biolog, and economics (Bole s Law, Ohm s Law, suppl and demand curves). A particularl significant application of hperbolas is found in the navigation sstems developed in World Wars I and II (see Eercise 5). Notice that the definition of a hperbola is similar to that of an ellipse; the onl change is that the sum of distances has become a difference of distances. In fact, the derivation of the equation of a hperbola is also similar to the one given earlier for an ellipse. It is left as Eercise 5 to show that when the foci are on the -ais at c, and the difference of distances is, then the equation of the hperbola is 6 PF PF a a b (_a, ) b =_ a (_c, ) FIGURE - = b = a (a, ) (c, ) where c a b. Notice that the -intercepts are again a and the points a, and a, are the vertices of the hperbola. But if we put in Equation 6 we get b, which is impossible, so there is no -intercept. The hperbola is smmetric with respect to both aes. To analze the hperbola further, we look at Equation 6 and obtain a b s a This shows that a, so. Therefore, we have a or a. This means that the hperbola consists of two parts, called its branches. When we draw a hperbola it is useful to first draw its asmptotes,which are the dashed lines ba and ba shown in Figure. Both branches of the hperbola approach the asmptotes; that is, the come arbitraril close to the asmptotes. a =_ b (, c) a = b 7 The hperbola a b has foci c,, where c a b, vertices a,, and asmptotes ba. (, a) (, _a) If the foci of a hperbola are on the -ais, then b reversing the roles of and we obtain the following information, which is illustrated in Figure. Thomson Brooks-Cole copright 7 FIGURE - = (, _c) 8 The hperbola a b has foci, c, where c a b, vertices, a, and asmptotes ab.

5 REVIEW OF CONIC SECTIONS 5 (_5, ) =_ 4 (_4, ) (4, ) = 4 (5, ) EXAMPLE 4 Find the foci and asmptotes of the hperbola and sketch its graph. SOLUTION If we divide both sides of the equation b 44, it becomes 6 9 FIGURE =44 which is of the form given in (7) with a 4 and b. Since c 6 9 5, the foci are 5,. The asmptotes are the lines and 4 4. The graph is shown in Figure 4. EXAMPLE 5 Find the foci and equation of the hperbola with vertices, and asmptote. SOLUTION From (8) and the given information, we see that a and ab. Thus, b a and c a b 5 4. The foci are (, s5) and the equation of the hperbola is 4 SHIFTED CONICS We shift conics b taking the standard equations (), (), (4), (5), (7), and (8) and replacing and b h and k. EXAMPLE 6 Find an equation of the ellipse with foci,, 4, and vertices,, 5,. SOLUTION The major ais is the line segment that joins the vertices,, 5, and has length 4, so a. The distance between the foci is, so c. Thus, b a c. Since the center of the ellipse is,, we replace and in (4) b and to obtain 4 as the equation of the ellipse. EXAMPLE 7 Sketch the conic -=_ (-4) and find its foci SOLUTION We complete the squares as follows: Thomson Brooks-Cole copright 7 (4, 4) (4, ) (4, _) -= (-4) FIGURE = This is in the form (8) ecept that and are replaced b 4 and. Thus, a 9, b 4, and c. The hperbola is shifted four units to the right and one unit upward. The foci are (4, s) and (4, s) and the vertices are 4, 4 and 4,. The asmptotes are 4. The hperbola is sketched in Figure 5.

6 6 REVIEW OF CONIC SECTIONS EXERCISES 8 Find the verte, focus, and directri of the parabola and sketch its graph Find an equation of the parabola. Then find the focus and directri Find the vertices and foci of the ellipse and sketch its graph A 5 6 Click here for answers. _ Find an equation of the ellipse. Then find its foci. S Click here for solutions Identif the tpe of conic section whose equation is given and find the vertices and foci Find an equation for the conic that satisfies the given conditions.. Parabola, verte,, focus,. Parabola, verte,, directri 5. Parabola, focus 4,, directri 4. Parabola, focus, 6, verte, 5. Parabola, verte,, ais the -ais, passing through (, 4) 6. Parabola, vertical ais, passing through,,,, and, 9 7. Ellipse, foci,, vertices 5, 8. Ellipse, foci, 5, vertices, 9. Ellipse, foci,,, 6 vertices,,, 8 4. Ellipse, foci,, 8,, verte 9, 4. Ellipse, center,, focus,, verte 5, 4. Ellipse, foci,, passing through, 4. Hperbola, foci,, vertices, 44. Hperbola, foci 6,, vertices 4, Hperbola, foci, and 7,, vertices, and 6, 46. Hperbola, foci, and, 8, vertices, and, Hperbola, vertices,, asmptotes 48. Hperbola, foci, and 6,, asmptotes and 6 Thomson Brooks-Cole copright 7 9 Find the vertices, foci, and asmptotes of the hperbola and sketch its graph The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo spacecraft was placed in an elliptical lunar orbit with perilune altitude km and apolune altitude 4 km (above the moon). Find an equation of this ellipse if the radius of the moon is 78 km and the center of the moon is at one focus.

7 REVIEW OF CONIC SECTIONS 7 5. A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is cm. (a) Find an equation of the parabola. (b) Find the diameter of the opening CD, cm from the verte. V A B 5 cm cm F 5 cm 5. In the LORAN (LOng RAnge Navigation) radio navigation sstem, two radio stations located at A and B transmit simultaneous signals to a ship or an aircraft located at P. The onboard computer converts the time difference in receiving these signals into a distance difference PA PB, and this, according to the definition of a hperbola, locates the ship or aircraft on one branch of a hperbola (see the figure). Suppose that station B is located 4 mi due east of station A on a coastline. A ship received the signal from B microseconds (s) before it received the signal from A. (a) Assuming that radio signals travel at a speed of 98 fts, find an equation of the hperbola on which the ship lies. (b) If the ship is due north of B, how far off the coastline is the ship? C D 56. (a) Show that the equation of the tangent line to the parabola 4p at the point, can be written as p. (b) What is the -intercept of this tangent line? Use this fact to draw the tangent line. 57. Use Simpson s Rule with n to estimate the length of the ellipse The planet Pluto travels in an elliptical orbit around the Sun (at one focus). The length of the major ais is.8 km and the length of the minor ais is.4 km. Use Simpson s Rule with n to estimate the distance traveled b the planet during one complete orbit around the Sun. 59. Let P, be a point on the ellipse a b with foci F and F and let and be the angles between the lines PF, PF and the ellipse as in the figure. Prove that. This eplains how whispering galleries and lithotrips work. Sound coming from one focus is reflected and passes through the other focus. [Hint: Use the formula to show that tan tan. See Challenge Problem..] å P(, ) F F + = m m tan m m A coastline 4 mi sending stations P B 6. Let P, be a point on the hperbola a b with foci F and F and let and be the angles between the lines PF, PF and the hperbola as shown in the figure. Prove that. (This is the reflection propert of the hperbola. It shows that light aimed at a focus F of a hperbolic mirror is reflected toward the other focus F.) 5. Use the definition of a hperbola to derive Equation 6 for a hperbola with foci c, and vertices a,. å P 5. Show that the function defined b the upper branch of the hperbola a b is concave upward. F F 54. Find an equation for the ellipse with foci, and, and major ais of length Determine the tpe of curve represented b the equation k k 6 P Thomson Brooks-Cole copright 7 in each of the following cases: (a) k 6, (b) k 6, and (c) k. (d) Show that all the curves in parts (a) and (b) have the same foci, no matter what the value of k is. F F

8 8 REVIEW OF CONIC SECTIONS ANSWERS S.,,, 8.,,,, Click here for solutions. ( 8, ), 8., 4, (, s) 4. ( 5, ), 4 _ ( s, ) =_ 8.,, (, 6), 6 4.,,,, = 6, _ 6 _4 5.,, (, s5) 6., and 5,, (s, ) (, ) (,_) 5.,,, 5, 6., 5, ( 5 4, 5), 4 (_, 5) = 7.,foci (, s5) ,foci ( s5, ) ,,,,., 4, (, s), 5 5 = 7.,, 5,, 8.,, (, 5 8 ), 7 8 (_5, _) (_, _).,, (, s),.,, (s, ), = = Thomson Brooks-Cole copright 7 9.,focus ( 4, ), directri 4.,focus (,, directri 5 ).,,,.,,, 6 _ œ 5 _œ 5 _. ( s6, ), ( s5, ), (s6) (, ) {+œ 5, }

9 REVIEW OF CONIC SECTIONS , 5 and, 5, , 5 and, 5, Parabola,,, (, 4) 6. Hperbola,,, (s, ) 7. Ellipse, (s, ),, 8. Parabola,, 4, (, 4) 9. Hperbola,,,, ; (, s5). Ellipse, (, ), (, s ) s7 s ,76,6,75,96 5. (a) p 5, (b) s 5. (a) (b) 48 mi,5,65,9, (a) Ellipse (b) Hperbola (c) No curve 56. (b) km Thomson Brooks-Cole copright 7

10 REVIEW OF CONIC SECTIONS SOLUTIONS. = =. 4p =,sop = 8.The verte is (, ),thefocusis 8,,andthe directri is = = = 4. 4p = 4,so p =. Theverteis(, ),thefocusis(, ), and the directri is =.. 4 = = 4. 4p = 4,so p =.Theverteis(, ),thefocusis 6, 6, and the directri is = =. 4p =,sop =.Theverteis(, ), the focus is (, ), and the directri is =. 5. ( +) =8( ). 4p =8,sop =.The verte is (, ),thefocusis(, 5),andthe directri is =. 6. =( +5). 4p =,sop = 4.Theverteis (, 5),thefocusis 5 4, 5, and the directri is = 4. Thomson Brooks-Cole copright 7

11 REVIEW OF CONIC SECTIONS = + += 4 ( +) = ( +). 4p =,sop =. The verte is (, ),thefocusis( 5, ),and the directri is = =6 = 6 ( 6 +9)= ( ) = + ( ) = ( +). 4p =,sop =.Theverteis(, ),thefocus 8 is, 5 8, and the directri is = The equation has the form =4p, wherep<. Since the parabola passes through (, ), wehave =4p( ),so4p = andanequationis = or =. 4p =,sop = 4 and the focus is 4, while the directri is = 4.. The verte is (, ), so the equation is of the form ( ) =4p( +),wherep>. The point (, ) is on the parabola, so 4=4p() and 4p =. Thus, an equation is ( ) =( +). 4p =,sop = and the focus is, while the directri is = = a = 9=, b = 5, c = a b = 9 5=. The ellipse is centered at (, ), with vertices at (±, ). The foci are (±, ) = a = =, b = 64 = 8, c = a b = 64 = 6. The ellipse is centered at (, ), with vertices at (, ±). The foci are (, ±6). Thomson Brooks-Cole copright 7

12 REVIEW OF CONIC SECTIONS. 4 + = = a = 6 = 4, b = 4=, c = a b = 6 4=.Theellipseis centered at (, ), with vertices at (, ±4). The foci are, ± =5 5/4 + = 5 a = = 5, b = =, 4 c = a b 5 = = =.The 4 4 ellipse is centered at (, ), with vertices at ± 5,. The foci are ±., =7 9( +)+4 =7+9 9( ) +4 ( ) =6 + = a =, b =, 4 9 c = 5 center (, ), vertices (, ±),foci, ± = ( + +)= 7+9+ ( ) +( +) =4 ( ) ( +) + = a =, b = =c center 4 (, ), vertices (, ) and (5, ),foci ±, 7. The center is (, ), a =,andb =, so an equation is =. c = a b = 5, so the foci are, ± The ellipse is centered at (, ), witha =and b =.Anequationis c = a b = 5,sothefociare ± 5,. ( ) 9 + ( ) 4 =. Thomson Brooks-Cole copright = a =, b =5, c = = center (, ), vertices (±, ),foci(±, ), asmptotes = ± 5. Note: It is helpful to draw a a-b-b rectangle whose center is the center of the hperbola. The asmptotes are the etended diagonals of the rectangle.

13 REVIEW OF CONIC SECTIONS. 6 = a =4, b =6, 6 c = a + b = = 5 =. The center is (, ), the vertices are (, ±4),thefociare, ±,andthe asmptotes are the lines = ± a b = ±.. =4 4 4 = a = 4==b, c = 4+4= center (, ), vertices (, ±), foci, ±, asmptotes = ±. 9 4 =6 4 9 = a = 4=, b = 9=, c = 4+9= center (, ), vertices (±, ),foci ±,,asmptotes = ±. 4 + = 8 ( +) ( 4 +4)= 8+ ( ) ( ) = 8 ( ) 6 ( ) 9 = a = 6, b =, c = 5 center (, ), vertices ± 6,,foci ± 5,,asmptotes =± 6 ( ) 6 or =± ( ) = 5 6( +4 +4) 9( + +5)= ( +) 9( +5) ( +) ( +5) =44 = 9 6 a =, b =4, c =5 center (, 5), vertices ( 5, 5) and (, 5),foci( 7, 5) and (, 5), asmptotes +5=± 4 ( +) Thomson Brooks-Cole copright 7 5. = + =( +).Thisisanequationofaparabola with 4p =,sop = 4.Theverteis(, ) and the focus is, 4.

14 4 REVIEW OF CONIC SECTIONS 6. = + =.Thisisanequationofahperbola with vertices (±, ). The foci are at ± +, = ±,. 7. =4 + 4 = +( +)= +( ) = ( ) + =. This is an equation of an ellipse with vertices at ±,. The foci are at ±, =(±, ) = =6 ( 4) =6.Thisisanequationofaparabola with 4p =6,sop =.Theverteis(, 4) and the focus is, = =4 +4 ( +) 4 =4 an equation of a hperbola with vertices (, ± ) = (, ) and (, ). The foci are at, ± 4+ =, ± 5. ( +) 4 =.Thisis = = = + + =.This /4 is an equation of an ellipse with vertices, ± =, ±.Thefociareat, ± =, ± 4 /.. Theparabolawithverte(, ) and focus (, ) opens downward and has p =, so its equation is =4p = 8.. The parabola with verte (, ) and directri = 5 opens to the right and has p =6, so its equation is =4p( ) = 4( ).. The distance from the focus ( 4, ) to the directri =is ( 4) = 6, so the distance from the focus to the verte is (6) = and the verte is (, ). Since the focus is to the left of the verte, p =. An equation is =4p( +) = ( +). 4. The distance from the focus (, 6) to the verte (, ) is 6 =4. Since the focus is above the verte, p =4.An equation is ( ) =4p( ) ( ) = 6( ). 5. The parabola must have equation =4p,so( 4) =4p() p =4 =6. 6. Vertical ais ( h) =4p( k). Substituting (, ) and (, ) gives ( h) =4p( k) and ( h) =4p( k) ( h) =( h) 4+4h + h = h h = =4p( k). Substituting (, 9) gives [ ( )] =4p(9 k) 4=4p(9 k). Solving for p from these equations gives p = 4( k) = 9 k +4 +=. 4( k) =9 k k = p = ( +) = ( ) 8 Thomson Brooks-Cole copright 7 7. The ellipse with foci (±, ) and vertices (±5, ) has center (, ) and a horizontal major ais, with a =5and c =,sob = a c =.Anequationis 5 + =.

15 REVIEW OF CONIC SECTIONS 5 8. The ellipse with foci (, ±5) and vertices (, ±) has center (, ) and a vertical major ais, with c =5and a =,sob = a c =.Anequationis =. 9. Since the vertices are (, ) and (, 8), the ellipse has center (, 4) with a vertical ais and a =4. The foci at (, ) and (, 6) are units from the center, so c =and b = a c = 4 =. An equation is ( ) ( 4) + = ( 4) + =. b a 6 4. Since the foci are (, ) and (8, ), the ellipse has center (4, ) with a horizontal ais and c =4.Theverte (9, ) is 5 units from the center, so a =5and b = a c = 5 4 = 9.Anequationis ( 4) ( +) + = a b ( 4) 5 + ( +) 9 =. 4. Center (, ), c =, a = b = 5 9 ( ) + 5 ( ) = 4. Center (, ), c =, major ais horizontal a + b =and b = a c = a 4. Since the ellipse passes through (, ),wehavea = PF + PF = 7 + a = 9+ 7 and b = + 7,sothe ellipse has equation =. 4. Center (, ), vertical ais, c =, a = b = 8= 8 = 44. Center (, ), horizontal ais, c =6, a =4 b = 5 6 = 45. Center (4, ), horizontal ais, c =, a = b = 5 4 ( 4) 5 ( ) = 46. Center (, ), vertical ais, c =5, a = b =4 9 ( ) 6 ( ) = 47. Center (, ), horizontal ais, a =, b a = b =6 9 6 = 48. Center (4, ), horizontal ais, asmptotes = ±( 4) c =, b/a = a = b c =4=a + b =a a = ( 4) ( ) = 49. In Figure 8, we see that the point on the ellipse closest to a focus is the closer verte (which is a distance a c from it) while the farthest point is the other verte (at a distance of a + c). So for this lunar orbit, (a c)+(a + c) =a = (78 + ) + (78 + 4),ora = 94;and(a + c) (a c) =c = 4, or c =. Thus, b = a c =,75,96,andtheequationis,76,6 +,75,96 =. 5. (a) Choose V to be the origin, with -ais through V and F. Then F is (p,), A is (p, 5), so substituting A into the equation =4p gives 5 = 4p so p = 5 and =. (b) = = CD = Thomson Brooks-Cole copright 7 5. (a) Set up the coordinate sstem so that A is (, ) and B is (, ). PA PB = ()(98) =,76, ft = 45 b = c a =,9,75,5,65,9,75 =. 5 mi =a a =,andc = so (b) Due north of B = ()(),5,65,575 = = 48 mi,9,75 59

16 6 REVIEW OF CONIC SECTIONS 5. PF PF = ±a ( + c) + ( c) + = ±a ( + c) + = ( c) + ± a ( + c) + =( c) + +4a ± 4a ( c) + 4c 4a = ±4a ( c) + c a c + a 4 = a c + c + c a a = a c a b a = a b (where b = c a ) a b = 5. The function whose graph is the upper branch of this hperbola is concave upward. The function is = f() =a + b = a b + b,so = a b b + / and = a b b + / b + / = ab b + / > for all,andsof is concave upward. 54. We can follow eactl the same sequence of steps as in the derivation of Formula 4, ecept we use the points (, ) and (, ) in the distance formula (first equation of that derivation) so ( ) +( ) + ( +) +( +) =4will lead (after moving the second term to the right, squaring, and simplifing) to ( +) +( +) = + +4, which, after squaring and simplifing again, leads to + = (a) If k>6, thenk 6 >, and k + =is an ellipse since it is the sum of two squares on the k 6 left side. (b) If <k<6,thenk 6 <,and k + =is a hperbola since it is the difference of two squares k 6 on the left side. (c) If k<,thenk 6 <, andthereisno curve since the left side is the sum of two negative terms, which cannot equal. (d) In case (a), a = k, b = k 6,andc = a b =6, so the foci are at (±4, ). Incase(b),k 6 <,so a = k, b =6 k,andc = a + b =6, and so again the foci are at (±4, ). 56. (a) =4p =4p = p, so the tangent line is = p ( ) =p( ) 4p =p p =p( + ). (b) The -intercept is. 57. Use the parametrization = cos t, = sin t, t π to get L =4 π/ (d/dt) +(d/dt) dt =4 π/ 4sin t +cos tdt=4 π/ sin t +dt Thomson Brooks-Cole copright 7 Using Simpson s Rule with n =, t = π/ = π,andf(t) = sin t +,weget L 4 π f() + 4f π +f π + +f 8π +4f 9π + f π 9.69

17 REVIEW OF CONIC SECTIONS The length of the major ais is a, soa = (.8 )= The length of the minor ais is b,so b = (.4 )= An equation of the ellipse is a + b =,orconvertingintoparametric equations, = a cos θ and = b sin θ. So L =4 π/ (d/dθ) +(d/dθ) dθ =4 π/ a sin θ + b cos θdθ Using Simpson s Rule with n =, θ = π/ = π,andf(θ) = a sin θ + b cos θ,weget L 4 S =4 π.64 km f() + 4f π +f π + +f 8π +4f 9π + f π 59. a + = b a + = = b b a ( 6= ). Thus, the slope of the tangent line at P is b. TheslopeofF P is a + c and of FP is. B the formula from Problems Plus, we have c tan α = + c + b a b a ( + c) = a + b ( + c) a ( + c) b = a b + b c c + a c using b + a = a b and a b = c = b (c + a ) c (c + a ) = b c and tan β = c b a b a ( c) So α = β. 6. The slopes of the line segments F P and F P are implicitl, a = = b b a Problems Plus, and tan α = + = a b ( c) a ( c) b = a b + b c c a c = b (c a ) c (c a ) = b c b a + c b a ( + c) = b (c + a ) c (c + a ) + c and c,wherep is (, ). Differentiating the slope of the tangent at P is b a, so b the formula from = b ( + c) a a ( + c)+b using /a /b = and a + b = c = b c Thomson Brooks-Cole copright 7 So α = β. tan β = + b a + c b a ( c) = b ( c)+a a ( c)+b = b (c a ) c (c a ) = b c

### THE PARABOLA 13.2. section

698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.

### 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

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,

### SECTION 9-1 Conic Sections; Parabola

66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships

### Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent

### Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0 P(X, Y) X

Rotation of Aes For a discussion of conic sections, see Appendi. In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the general second-degree

### THE PARABOLA section. Developing the Equation

80 (-0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or

### 9.5 CALCULUS AND POLAR COORDINATES

smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet

### Name Class Date. Deriving the Equation of a Parabola

Name Class Date Parabolas Going Deeper Essential question: What are the defining features of a parabola? Like the circle, the ellipse, and the hperbola, the parabola can be defined in terms of distance.

### 10.2. Introduction to Conics: Parabolas. Conics. What you should learn. Why you should learn it

3330_00.qd /8/05 9:00 AM Page 735 Section 0. Introduction to Conics: Parabolas 735 0. Introduction to Conics: Parabolas What ou should learn Recognize a conic as the intersection of a plane and a double-napped

### REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

### Warm-Up y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.

CST/CAHSEE: Warm-Up Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and

### 17.1. Conic Sections. Introduction. Prerequisites. Learning Outcomes. Learning Style

Conic Sections 17.1 Introduction The conic sections (or conics) - the ellipse, the parabola and the hperbola - pla an important role both in mathematics and in the application of mathematics to engineering.

### Section 7-2 Ellipse. Definition of an Ellipse The following is a coordinate-free definition of an ellipse: DEFINITION

7- Ellipse 3. Signal Light. A signal light on a ship is a spotlight with parallel reflected light ras (see the figure). Suppose the parabolic reflector is 1 inches in diameter and the light source is located

### Additional Topics in Analytic Geometry

bar1969_ch11_961-984.qd 17/1/08 11:43 PM Page 961 Pinnacle ju111:venus:mhia06:mhia06:student EDITION:CH 11: CHAPTER Additional Topics in Analtic Geometr C ANALYTIC geometr is the stud of geometric objects

### conics and polar coordinates

17 Contents nts conics and polar coordinates 1. Conic sections 2. Polar coordinates 3. Parametric curves Learning outcomes In this workbook ou will learn about some of the most important curves in the

### Conics and Polar Coordinates

Contents 17 Conics and Polar Coordinates 17.1 Conic Sections 2 17.2 Polar Coordinates 23 17.3 Parametric Curves 33 Learning outcomes In this Workbook ou will learn about some of the most important curves

### 7.3 Parabolas. 7.3 Parabolas 505

7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of

### Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304

Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

### Chapter 7: Eigenvalues and Eigenvectors 16

Chapter 7: Eigenvalues and Eigenvectors 6 SECION G Sketching Conics B the end of this section ou will be able to recognise equations of different tpes of conics complete the square and use this to sketch

### 8.7 The Parabola. PF = PD The fixed point F is called the focus. The fixed line l is called the directrix.

8.7 The Parabola The Hubble Space Telescope orbits the Earth at an altitude of approimatel 600 km. The telescope takes about ninet minutes to complete one orbit. Since it orbits above the Earth s atmosphere,

### 2.1 Three Dimensional Curves and Surfaces

. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

### Locus and the Parabola

Locus and the Parabola TERMINOLOGY Ais: A line around which a curve is reflected eg the ais of symmetry of a parabola Cartesian equation: An equation involving two variables and y Chord: An interval joining

### Section 10-5 Parametric Equations

88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

### A CLASSROOM NOTE ON PARABOLAS USING THE MIRAGE ILLUSION

A CLASSROOM NOTE ON PARABOLAS USING THE MIRAGE ILLUSION Abstract. The present work is intended as a classroom note on the topic of parabolas. We present several real world applications of parabolas, outline

### Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Algebra and coordinate geometry: Module 2. Coordinate geometry

1 Supporting Australian Mathematics Project 3 4 5 6 7 8 9 1 11 1 A guide for teachers Years 11 and 1 Algebra and coordinate geometr: Module Coordinate geometr Coordinate geometr A guide for teachers (Years

### Identifying second degree equations

Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

### Chapter 10: Topics in Analytic Geometry

Chapter 10: Topics in Analytic Geometry 10.1 Parabolas V In blue we see the parabola. It may be defined as the locus of points in the plane that a equidistant from a fixed point (F, the focus) and a fixed

### DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

### Answers (Anticipation Guide and Lesson 10-1)

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

### Quadratic Functions. MathsStart. Topic 3

MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE

8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

### Solving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots

### Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

### Calculus with Parametric Curves

Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function

### Polynomial and Rational Functions

Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

### C1: Coordinate geometry of straight lines

B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

### The Parabola. By: OpenStaxCollege

The Parabola By: OpenStaxCollege The Olympic torch concludes its journey around the world when it is used to light the Olympic cauldron during the opening ceremony. (credit: Ken Hackman, U.S. Air Force)

### Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal

Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### Exponential and Logarithmic Functions

Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

### Analyzing the Graph of a Function

SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

### Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

### Deriving the Standard-Form Equation of a Parabola

Name Class Date. Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Eplore Deriving the Standard-Form Equation of a

### Chapter 10: Analytic Geometry

10.1 Parabolas Chapter 10: Analytic Geometry We ve looked at parabolas before when talking about the graphs of quadratic functions. In this section, parabolas are discussed from a geometrical viewpoint.

### 6.3 Parametric Equations and Motion

SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh

Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This

### Conics. Find the equation of the parabola which has its vertex at the origin and its focus at point F in the following cases.

Conics 1 Find the equation of the parabola which has its vertex at the origin and its focus at point F in the following cases. a) F(, 0) b) F(0,-4) c) F(-3,0) d) F(0, 5) In the Cartesian plane, represent

### The Parabola. The Parabola in Terms of a Locus of Points

The Parabola Appolonius of Perga (5 B.C.) discovered that b intersecting a right circular cone with a plane slanted the same as the side of the cone, (formall, when it is parallel to the slant height),

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### 7-2 Ellipses and Circles

Graph the ellipse given by each equation. 1. + = 1 The ellipse is in standard form, where h = 2, k = 0, a = or 7, b = or 3, and c = or about 6.3. The orientation is vertical because the y term contains

### INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

### A Summary of Curve Sketching. Analyzing the Graph of a Function

0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph

CHALLENGE PROBLEMS CHALLENGE PROBLEMS: CHAPTER 0 A Click here for answers. S Click here for solutions. m m m FIGURE FOR PROBLEM N W F FIGURE FOR PROBLEM 5. Each edge of a cubical bo has length m. The bo

0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

### ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

### x Q.7 All points on the curve y 2 4a

C X,Y CONIC SECTION PRACTICE SHEET Q. If on a given base, a triangle be described such that the sum of the tangents of the base angles is a constant, then the locus of the verte is : a circle (B) a parabola

### Change of Coordinates in Two Dimensions

CHAPTER 5 Change of Coordinates in Two Dimensions Suppose that E is an ellipse centered at the origin. If the major and minor aes are horiontal and vertical, as in figure 5., then the equation of the ellipse

### Graphing and transforming functions

Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6

### Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

### Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

### 10-5 Parabolas. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

10-5 Parabolas Warm Up Lesson Presentation Lesson Quiz 2 Warm Up 1. Given, solve for p when c = Find each distance. 2. from (0, 2) to (12, 7) 13 3. from the line y = 6 to (12, 7) 13 Objectives Write the

Section 3.1 Quadratic Functions 1 3.1 Quadratic Functions Functions Let s quickl review again the definition of a function. Definition 1 A relation is a function if and onl if each object in its domain

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### June 2011 PURDUE UNIVERSITY Study Guide for the Credit Exams in Single Variable Calculus (MA 165, 166)

June PURDUE UNIVERSITY Stud Guide for the Credit Eams in Single Variable Calculus (MA 65, 66) Eam and Eam cover respectivel the material in Purdue s courses MA 65 (MA 6) and MA 66 (MA 6). These are two

### decide, when given the eccentricity of a conic, whether the conic is an ellipse, a parabola or a hyperbola;

Conic sections In this unit we study the conic sections. These are the curves obtained when a cone is cut by a plane. We find the equations of one of these curves, the parabola, by using an alternative

### Name: Class: Date: Conics Multiple Choice Post-Test. Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: Conics Multiple Choice Post-Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1 Graph the equation x = 4(y 2) 2 + 1. Then describe the

### 8 Graphs of Quadratic Expressions: The Parabola

8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be

Algebra Module A7 The Parabola Copright This publication The Northern Alberta Institute of Technolog. All Rights Reserved. LAST REVISED December, The Parabola Statement of Prerequisite Skills Complete

### 9-3 Polar and Rectangular Forms of Equations

9-3 Polar and Rectangular Forms of Equations Find the rectangular coordinates for each point with the given polar coordinates Round to the nearest hundredth, if necessary are The rectangular coordinates

### Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Functions: Module 4. Quadratics

1 Supporting Australian Mathematics Project 2 3 4 5 6 7 8 9 1 11 12 A guide for teachers Years 11 and 12 Functions: Module 4 Quadratics Quadratics A guide for teachers (Years 11 12) Principal author: Peter

### Astromechanics Two-Body Problem (Cont)

5. Orbit Characteristics Astromechanics Two-Body Problem (Cont) We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the

### Rotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012

Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to

### ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

### Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

### 13.4 THE CROSS PRODUCT

710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product

### OpenStax-CNX module: m The Parabola. OpenStax College. Abstract

OpenStax-CNX module: m49440 1 The Parabola OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will: Abstract Graph

### Study Guide and Review

For each equation, identify the vertex, focus, axis of symmetry, and directrix. Then graph the 11. (x + 3) 2 = 12(y + 2) (x + 3) 2 = 12(y + 2) The equation is in standard form and the squared term is x,

### Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

### HI-RES STILL TO BE SUPPLIED

1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

### Archimedes quadrature of the parabola and the method of exhaustion

JOHN OTT COLLEGE rchimedes quadrature of the parabola and the method of ehaustion CLCULUS II SCIENCE) Carefull stud the tet below and attempt the eercises at the end. You will be evaluated on this material

### MATH122 Final Exam Review If c represents the length of the hypotenuse, solve the right triangle with a = 6 and B = 71.

MATH1 Final Exam Review 5. 1. If c represents the length of the hypotenuse, solve the right triangle with a = 6 and B = 71. 5.1. If 8 secθ = and θ is an acute angle, find sinθ exactly. 3 5.1 3a. Convert

### The Distance Formula and the Circle

10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

### Name: x 2 + 4x + 4 = -6y p = -6 (opens down) p = -3/2 = (x + 2) 2 = -6y + 6. (x + 2) 2 = -6(y - 1) standard form: (-2, 1) Vertex:

Write the equation of the parabola in standard form and sketch the graph of the parabola, labeling all points, and using the focal width as a guide for the width of the parabola. Find the vertex, focus,

### Essential Question: What is the relationship among the focus, directrix, and vertex of a parabola?

Name Period Date: Topic: 9-3 Parabolas Essential Question: What is the relationship among the focus, directrix, and vertex of a parabola? Standard: G-GPE.2 Objective: Derive the equation of a parabola

### Multivariable Calculus MA22S1. Dr Stephen Britton

Multivariable Calculus MAS Dr Stephen Britton September 3 Introduction The aim of this course to introduce you to the basics of multivariable calculus. The early part is dedicated to discussing curves

### 1. x = 2 t, y = 1 3t, z = 3 2t. 4. x = 2+t, y = 1 3t, z = 3+2t. 5. x = 1 2t, y = 3 t, z = 2 3t. x = 1+2t, y = 3 t, z = 2+3t

Version 1 Homework 4 gri (11111) 1 This print-out should have 1 questions. Multiple-choice questions ma continue on the net column or page find all choices before answering. CalC13e11b 1 1. points A line

### Attributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest y-value of a function. Minimum value the least

- Attributes and Transformations of Quadratic Functions TEKS FCUS TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening.

### Algebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2

1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3-dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect

### SL Calculus Practice Problems

Alei - Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram

### North Carolina Community College System Diagnostic and Placement Test Sample Questions

North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College

### Unit 5 Conic Sections

Mathematics III Unit 5 nd Edition Mathematics III Frameworks Student Edition Unit 5 Conic Sections nd Edition June, 00 June, 00 Copyright 00 All Rights Reserved Unit : Page of 47 Mathematics III Unit 5

### Chapter 12 Notes, Calculus I with Precalculus 3e Larson/Edwards. Contents Parabolas Ellipse Hyperbola...

Contents 1.1 Parabolas.............................................. 1. Ellipse................................................ 6 1.3 Hyperbola.............................................. 10 1 1.1 Parabolas

### 2 Analysis of Graphs of

ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,