MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Transcription

1 Problems to look over Ch and Section 1. Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the ellipse and locate the foci. 1) x y 2 = 1 1) A) foci at (2 10, 0) and (-2 10, 0) B) foci at (0, 2 10) and (0, -2 10) C) foci at (0, 55) and (0, - 55) D) foci at ( 55, 0) and (- 55, 0) 1

2 Find the standard form of the equation of the ellipse and give the location of its foci. 2) 2) Answer: C A) x y 2 64 = 1 foci at (-, 0) and (, 0) C) x y 2 64 = 1 foci at (- 17, 0) and ( 17, 0) B) x y 2 81 = 1 foci at (- 17, 0) and ( 17, 0) D) x y 2 64 = 1 foci at (- 17, 0) and ( 17, 0) 2

3 Graph the ellipse. 3) (x - 1) 2 + (y - 2) 2 4 = 1 3) A) B) C) D) Find the vertices and locate the foci for the hyperbola whose equation is given. 4) x y 2 = 1 4) 16 A) vertices: (-4, 0), (4, 0) foci: (- 65, 0), ( 65, 0) C) vertices: (0, -7), (0, 7) foci: (- 65, 0), ( 65, 0) B) vertices: (-7, 0), (7, 0) foci: (- 65, 0), ( 65, 0) D) vertices: (-7, 0), (7, 0) foci: (-4, 0), (4, 0) Find the standard form of the equation of the hyperbola satisfying the given conditions. 5) Foci: (-, 0), (, 0); vertices: (-6, 0), (6, 0) 5) A) x y 2 45 = 1 B) x y 2 81 = 1 C) y x 2 81 = 1 D) y x 2 45 = 1 3

4 6) Use vertices and asymptotes to graph the hyperbola. Find the equations of the asymptotes. 6) x2 - y 2 25 = 1 Answer: D A) Asymptotes: y = ± 3 5 x B) Asymptotes: y = ± 5 3 x C) Asymptotes: y = ± 3 5 x D) Asymptotes: y = ± 5 3 x Find the location of the center, vertices, and foci for the hyperbola described by the equation. 7) (x + 4) 2 - (y + 2) 2 = 1 Answer: D 7) 4 64 A) Center: (-4, -2); Vertices: (-6, 2) and (-2, 2); Foci: ( , 2) and ( , 2) B) Center: (-4, -2); Vertices: (-5, -2) and (-1, -2); Foci: ( , -1) and ( , -1) C) Center: (4, 2); Vertices: (2, 2) and (6, 2); Foci: (4-2 17, 2) and ( , 2) D) Center: (-4, -2); Vertices: (-6, -2) and (-2, -2); Foci: ( , -2) and ( , -2) Find the focus and directrix of the parabola with the given equation. 8) x2 = 8y 8) A) focus: (0, 2) directrix: y = -2 B) focus: (0, -2) directrix: x = -2 C) focus: (2, 0) directrix: y = 2 D) focus: (2, 0) directrix: x = 2 4

5 Find the standard form of the equation of the parabola using the information given. ) Focus: (22, 0); Directrix: x = -22 ) A) y2 = -88x B) y2 = 88x C) x2 = 88y D) y2 = 22x Find the vertex, focus, and directrix of the parabola with the given equation. 10) (y - 2)2 = 20(x - 3) Answer: C 10) A) vertex: (3, 2) focus: (-2, 2) directrix: x = 8 C) vertex: (3, 2) focus: (8, 2) directrix: x = -2 B) vertex: (-3, -2) focus: (2, -2) directrix: x = -8 D) vertex: (2, 3) focus: (7, 3) directrix: x = -3 Parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. 11) x = t3 + 1, y = - t4; t = 2 Answer: C 11) A) (17, -15) B) (17, 1) C) (, -7) D) (, 25) Use point plotting to graph the plane curve described by the given parametric equations. 12) x = 2t, y = t + 3; -2 t 3 12) A) B) C) D) 5

6 13) x = 5 sin t, y = 5 cos t; 0 t 2 13) A) B) C) D) Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations. 14) x = 3t, y = t + 3; -2 t 3 14) A) y = 1 3 x + 3; -6 x B) y = 1 x - 3; - < x < 3 C) y = x2 + 1; -2 x 2 D) y = -3x + 3; - < x < 15) x = t3 + 1, y = t3-1; -2 t 2 15) A) y = x3; -3 x 1 B) y = -x2; -4 x 4 C) y = x - 2; -7 x D) y = -x - 2; -7 x Eliminate the parameter. Write the resulting equation in standard form. 16) An ellipse: x = cos t, y = sin t 16) A) (x + 3) (y + 5) 2 = 1 B) (x - 3) 2 4 C) (x - 3)2 + (y - 5)2 = 1 D) (x - 5) 2 + (y - 5) 2 + (y - 3) 2 4 = 1 = 1 6

7 Find a set of parametric equations for the conic section or the line. 17) Circle: Center: (2, 3); Radius: 2 17) A) x = cos t; y = sin t B) x = sin t; y = cos t C) x = 2 + sin t; y = 3 + cos t D) x = t - 2; (y - 3)2 + t2 = 4 18) Ellipse: Center: (-4, -5); Vertices: 5 units above and below the center; Endpoints of Minor Axis: 2 18) units left and right of the center. A) x = 2-4 cos t, y = 5-5 sin t B) x = cos t, y = sin t C) x = -4-2 cos t, y = -5-5 sin t D) x = cos t, y = sin t Find a set of parametric equations for the rectangular equation. 1) y = 4x - 1 1) A) x = t; y = 4t - 1 B) y = 4t; 4x = t + 1 C) x = t; y = 4t2-1 D) x = t 4 ; y = t Find two sets of parametric equations for the given rectangular equation. 20) y = 3x + 6 Answer: C 20) A) x = 3t, y = t + 6; x = t 3, y = t + 6 B) x = t, y = 3t + 6; x = t, y = t C) x = t, y = 3t + 6; x = t, y = t + 6 D) x = t, y = 3t + 6; x = 3t, y = t Solve the problem. 21) Ron throws a ball straight up with an initial velocity of 40 feet per second from a height of 3 feet. 21) Find parametric equations that describe the motion of the ball as a function of time. How long is the ball in the air? When is the ball at its maximum height? What is the maximum height of the ball? A) x = 0; y = -16t2 + 40t + 3; sec; 1.25 sec; 25 feet C) x = 0; y = -16t2 + 40t + 3; sec; 1.25 sec; 2.85 feet B) x = 0; y = -16t2 + 40t + 3; sec; 1.25 sec; 28 feet D) x = 0; y = -16t2 + 40t + 3; sec; 1.25 sec; feet 7

8 Use a graphing utility to obtain the plane curve represented by the given parametric equations. 22) Cycloid: x = 2(t - sin t), y= 2(1 - cos t), 0 t 6 22) A) B) C) D) 8

9 Sketch the plane curve represented by the given parametric equations. Then use interval notation to give the relation's domain and range. 23) x = 2t, y = t2 + t ) A) Domain: (-, ); Range: [2.75, ) B) Domain: (-, ); Range: [2.75, ) C) Domain: (-, ); Range: - 1x, ) D) Domain: (-, ); Range: [3, ) Write the standard form of the equation of the circle with the given center and radius. 24) (7, 8); 24) A) (x - 7)2 + (y - 8)2 = 81 B) (x + 7)2 + (y + 8)2 = 81 C) (x + 8)2 + (y + 7)2 = D) (x - 8)2 + (y - 7)2 = Find the center and the radius of the circle. 25) (x + 7)2 + (y + 8)2 = 4 25) A) (7, 8), r = 4 B) (8, 7), r = 4 C) (-7, -8), r = 7 D) (-8, -7), r = 7 Complete the square and write the equation in standard form. Then give the center and radius of the circle. 26) x2 + 6x + + y2 + 16y + 64 = 64 26) A) (x + 3)2 + (y + 8)2 = 64 (-3, -8), r = 8 C) (x + 8)2 + (y + 3)2 = 64 (-8, -3), r = 8 B) (x + 3)2 + (y + 8)2 = 64 (3, 8), r = 64 D) (x + 8)2 + (y + 3)2 = 64 (8, 3), r = 64