Deck 13: Topics in Analytic Geometry

Question 1

Match the graph with its equation.

A)
B)
C)
D)
E)

Answer 1

Question 2

Find the standard form of the parabola with the given characteristic and vertex at the origin.vertical axis and passes through point

A)
B)
C)
D)
E)

Answer 2

Question 3

Find the standard form of the equation of the hyperbola with the given characteristics.vertices: (0, $\pm$ 2) foci: (0, $\pm$ 8)

A)
B)
C)
D)
E)

Answer 3

Question 4

Find the standard form of the equation of the ellipse with the following characteristics.foci: major axis of length: 20

A)
B)
C)
D)
E)

Question 5

Find the center and foci of the ellipse.

A)center: (5, -9) foci: (5, -15), (5, -3)
B)center: (-5, 9) foci: (-5, 3), (-5, 15)
C)center: (-5, 9) foci: (-11, 9), (1, 9)
D)center: (5, -9) foci: (-1, 9), (11, 9)
E)center: (-5, 9) foci: (-1, -9), (11, -9)

Question 6

Find the vertices and asymptotes of the hyperbola.

A)vertices: (0, $\pm$ 5) asymptote:
B)vertices: (0, $\pm$ 5) asymptote:
C)vertices: ( $\pm$ 5, 0) asymptote:
D)vertices: ( $\pm$ 5, 0) asymptote:
E)vertices: ( $\pm$ 5, 2) asymptote:

Question 7

Find the standard form of the parabola with the given characteristics.focus: (2, 9) vertex: (2, 4)

A)(x - 2)² = 20(y - 4)
B)(x + 2)² = 20(y + 4)
C)(x - 4)² = 20(y - 2)
D)(x + 4)² = 20(y + 2)
E)(x + 2)² = 5(y + 4)

Question 8

Find the standard form of the equation of the ellipse with the given characteristics.center: (3, -5) foci: (-2, -5), (8, -5)

A)
B)
C)
D)
E)

Question 9

Find the standard form of the parabola with the given characteristics.directrix: x = 9 vertex: (4, 6)

A)(x + 4)² = -20(y + 6)
B)(x - 4)² = -20(y - 6)
C)(x + 6)² = -20(y + 4)
D)(y + 6)² = -20(x + 4)
E)(y - 6)² = -20(x - 4)

Question 10

Find the standard form of the parabola with the given characteristic and vertex at the origin.directrix: x = 8

A)x² = -32y
B)x² = 32y
C)x² = 8y
D)y² = 8x
E)y² = -32x

Question 11

Find the standard form of the parabola with the given characteristics.focus: (8, -11) directrix: y = -1

A)(x - 8)² = -20(y + 6)
B)(x + 8)² = -20(y - 6)
C)(x + 6)² = -20(y - 8)
D)(x - 6)² = -20(y + 8)
E)(x + 8)² = -5(y - 6)

Question 12

Find the center and vertices of the ellipse. = 0

A)center: (-5, -9) vertices: (-8, -9), (-2, -9)
B)center: (9, 5) vertices: (8, 5), (10, 5)
C)center: (-9, -5) vertices: (-10, -5), (-8, -5)
D)center: (-9, -5) vertices: (-12, -5), (-6, -5)
E)center: (9, 5) vertices: (6, 5), (12, 5)

Question 13

Find the center and foci of the hyperbola.

A)center: (8, -4) foci: (8, -10), (8, 2)
B)center: (-8, 4) foci: (-8, -2), (-8, 10)
C)center: (4, -8) foci: (-2, -8), (10, -8)
D)center: (-8, 4) foci: (-14, 4), (-2, 4)
E)center: (-4, 8) foci: (-4, 2), (-4, 14)

Question 14

Match the graph with its equation.

A)
B)
C)
D)
E)

Question 15

Find the standard form of the equation of the ellipse with the given characteristics.vertices: (-1, 1), (-1, 17) minor axis of length 14

A)
B)
C)
D)
E)

Question 16

Find the standard form or the equation of the ellipse with vertices ( $\pm$ 7, 0) and eccentricity .

A)
B)
C)
D)
E)

Question 17

Find the vertex and focus of the parabola.

A)vertex: focus: (0, 0)
B)vertex: focus: (0, 0)
C)vertex: (0, 0) focus:
D)vertex: (0, 0) focus:
E)vertex: (0, 0) focus:

Question 18

Find the standard form of the parabola with the given characteristic and vertex at the origin.focus: (0, 9)

A)x² = 36y
B)x² = 9y
C)x² = -9y
D)y² = 36x
E)y² = 9x

Question 19

Match the graph with its equation.

A)
B)
C)
D)
E)

Question 20

Find the center and vertices of the hyperbola.

A)center: (2, 5) vertices: (2, 3), (2, 7)
B)center: (-2, -5) vertices: (-2, -7), (-2, -3)
C)center: (-2, -5) vertices: (-4, -5), (0, -5)
D)center: (2, 5) vertices: (0, 5), (4, 5)
E)center: (5, 2) vertices: (3, 2), (7, 2)

Question 21

Which answer is a polar form of the given rectangular equation?

A)
B)
C)
D)
E)

Question 22

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.6x² - y² - 8x + y + 8 = 0

A)ellipse
B)parabola
C)hyperbola
D)circle

Question 23

Which answer is a set of polar coordinates for the following rectangular coordinates? Answers are rounded to three decimal places, if necessary.

A)
B)
C)
D)
E)

Question 24

Which answer is a rectangular form of the given polar equation?

A)
B)
C)
D)
E)

Question 25

Find the standard form of the equation of the hyperbola with the given characteristics.vertices: (5, 0), (5, 4) foci: (5, -1), (5, 5)

A)
B)
C)
D)
E)

Question 26

Find the standard form of the equation of the hyperbola with the given characteristics.vertices: (0, 4), (4, 4) asymptotes: ,

A)
B)
C)
D)
E)

Question 27

Which group of polar coordinates contains only representations of the point shown in the graph?

A)
B)
C)
D)
E)

Question 28

Which set of parametric equations represents the following line or conic?Use and .Line: passes through

A)
B)
C)
D)
E)

Question 29

Find the standard form of the equation of the hyperbola with the given characteristics.foci: ( $\pm$ 5, 0) asymptotes:

A)
B)
C)
D)
E)

Question 30

Which answer is a rectangular form of the given polar equation?

A)
B)
C)
D)
E)

Question 31

Which set of parametric equations represents the following line or conic?Use and .Hyperbola: vertices ,foci

A)
B)
C)
D)
E)

Question 32

Match the parametric equations to the curve they represent and the corresponding rectangular equation. Adjust the domain of the resulting rectangular equation if necessary.

A)
B)
C)
D)
E)

Question 33

Which set of parametric equations represents the graph of the following rectangular equation using ?

A)
B)
C)
D)
E)

Question 34

Use a graphing utility to graph the curve represented by the parametric equations.

Question 35

Convert the point from polar coordinates to rectangular coordinates. Round answer to three decimal places if necessary.

A)
B)
C)
D)
E)

Question 36

Analyze the curves below and determine the way(s) the plane curves differ from each other. Make use of sketchs of the graphs to aid in your analysis.

Question 37

Which answer is a polar form of the given rectangular equation?

A)
B)
C)
D)
E)

Question 38

Which set of parametric equations represents the following line or conic?Use and .Ellipse: vertices ,foci

A)
B)
C)
D)
E)

Question 39

A projectile is launched from ground level at an angle of $\theta$ with the horizontal. The initial velocity is feet per second and the path of the projectile is modeled by the parametric equations and .Use a graphing utility to graph the paths of a projectile launched from ground level with the values given for $\theta$ and . Use the graph to approximate the maximum height and range of the projectile to the nearest foot. ,

A)maximum height $\approx$ 100 feet range $\approx$ 9 feet
B)maximum height $\approx$ 9 feet range $\approx$ 100 feet
C)maximum height $\approx$ 67 feet range $\approx$ 188 feet
D)maximum height $\approx$ 188 feet range $\approx$ 67 feet
E)maximum height $\approx$ 43 feet range $\approx$ 8 feet

Question 40

Match the graph to a set of parametric equations.

A)
B)
C)
D)
E)

Question 41

Write the polar equation of the conic for . Identify the conic.

A)ellipse:
B)ellipse:
C)parabola:
D)hyperbola:
E)hyperbola:

Question 42

Find a polar equation of the conic with the given characteristics and with one focus at the pole.Conic Eccentricity DirectrixParabola

A)
B)
C)
D)
E)

Question 43

Find any zeros of r on the interval .

A)zeros:
B)zeros:
C)zeros:
D)zeros:
E)zeros:

Question 44

Test for symmetry with respect to $\theta$ = $\pi$ /2, the polar axis, and the pole.

A)symmetric with respect to the pole
B)symmetric with respect to the polar axis
C)symmetric with respect to $\theta$ = $\pi$ /2
D)symmetric with respect to the pole and the polar axis
E)symmetric with respect to the pole and $\theta$ = $\pi$ /2

Question 45

Match the graph to its polar equation.

A)
B)
C)
D)
E)

Question 46

Find a polar equation of the conic with the given characteristics and with one focus at the pole.Conic VerticesHyperbola

A)
B)
C)
D)
E)

Question 47

If a new comet were discovered with an eccentricity of and whose length of the major axis of orbit is approximately 6.46 astronomical units, how close would the comet come to the sun (perihelion distance)? The perihelion distance is defined by the equation . Round answer to three decimal places, if necessary.

A)approximately 2.778 astronomical units
B)approximately 1.389 astronomical units
C)approximately 3.682 astronomical units
D)approximately 1.841 astronomical units
E)approximately 1.14 astronomical units

Question 48

Find a polar equation of the conic with the given characteristics and with one focus at the pole.Conic Eccentricity DirectrixHyperbola

A)
B)
C)
D)
E)

Question 49

Sketch the graph of the polar equation using symmetry, zeros, maximum r-values and any other additional points. Use either grid below for your graph, whichever is more convenient.

Question 50

Use a graphing utility to graph the polar equation. Describe the resulting graph.

A)rose curve with 5 petals, rotated 5 radians
B)rose curve with 5 petals, rotated 3 radians
C)lemniscate rotated 3 radians
D)limaçon rotated 3 radians
E)cardioid rotated 5 radians

Question 51

Find a polar equation of the conic with the given characteristics and with one focus at the pole.Conic Eccentricity DirectrixEllipse

A)
B)
C)
D)
E)

Question 52

Find an interval for $\theta$ for which the graph is traced only once.

A)
B)
C)
D)
E)

Question 53

Find a polar equation of the conic with the given characteristics and with one focus at the pole.Conic VerticesEllipse

A)
B)
C)
D)
E)

Question 54

Use a graphing utility to graph the polar equation and show that the indicated line is an asymptote.Name of Graph Polar Equation AsymptoteStrophoid Use either grid below for your graph, whichever is more convenient.

Question 55

Use a graphing utility to graph the rotated conic. Use either grid below for your graph, whichever is more convenient.

Question 56

Identify the conic and sketch its graph. Use either grid below for your graph, whichever is more convenient.

Question 57

Identify the type of polar graph.

A)
B)
C)
D)
E)

Question 58

Test for symmetry with respect to $\theta$ = $\pi$ /2, the polar axis, and the pole.

A)symmetric with respect to the pole
B)symmetric with respect to the polar axis
C)symmetric with respect to $\theta$ = $\pi$ /2
D)symmetric with respect to the pole and the polar axis
E)symmetric with respect to the pole and $\theta$ = $\pi$ /2

Question 59

Find the maximum value of on the interval .

A)maximum value of : r = 25 where
B)maximum value of : r = 25 where
C)maximum value of : r = 5 where
D)maximum value of : r = 5 where
E)maximum value of : r = 5 where

Question 60

Find a polar equation of the conic with the given characteristics and with the focus at the pole.Conic VertexParabola

A)
B)
C)
D)
E)