Deck 12: Conic Sections

Question 1

Find an equation for the hyperbola that has vertices
and asymptotes
.

Answer 1

Question 2

Find an equation for the ellipse with foci
, and major axis of length
.

Answer 2

Question 3

Find the lengths of the major and minor axes for the ellipse.

Answer 3

Length of the major axis is
, and minor is
.

Question 4

Find the foci of the hyperbola.

Question 5

Sketch the graph of the ellipse.

Question 6

Sketch the graph of the hyperbola.

Question 7

Complete the square to determine the type of curve represented by the equation.

Question 8

Find an equation for the parabola with vertex
and directrix
.

Question 9

Find an equation of the parabola whose graph is shown.

Question 10

Find the vertices and foci for the ellipse.

Question 11

Find the vertex, focus, and directrix of the parabola
.

Question 12

Find an equation for the conic whose graph is shown.

Question 13

Determine the $X Y$ - coordinates of $( 6,0 )$ if the axes are rotated through an angle $\phi = 60 ^ { \circ }$

Question 14

Sketch the graph of the parabola.

Question 15

A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 ft from the cannon and the highest point it reaches is 2400 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.

Question 16

Complete the square to determine the type of curve represented by the equation.

Question 17

Find the vertices of the hyperbola.

Question 18

Find an equation of the parabola whose graph is shown.

Question 19

Find an equation for parabola with vertex
and directrix
.

Question 20

Sketch the graph of the parabola.

Question 21

Find an equation for the conic whose graph is shown.

Question 22

Find an equation for the conic whose graph is shown.

Question 23

Complete the square to determine the type of curve represented by the equation.

Question 24

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola.

Question 25

Write a polar equation of an ellipse with eccentricity $0.3$ and directrix $y = - 7$

A) $r= \frac { 7 } { 10 - 3 \sin \theta }$

B) $r= \frac { 2 } { 1 - 3 \sin \theta }$

C) $r= \frac { 21 } { 10 + 3 \sin \theta }$

D) $r = \frac { 21 } { 10 - 3 \sin \theta }$

E) $r= \frac { 21 } { 10 - 3 \cos \theta }$

Question 26

Sketch the graph of the parabola.

Question 27

Write a polar equation of an ellipse with eccentricity $0.3$
and directrix $y = - 7$
.

Question 28

Find the eccentricity and identify the conic given by $r = \frac { 2 } { 1 - \cos \theta }$
.

Question 29

Find the vertices and foci for the ellipse.

Question 30

Sketch the graph of the parabola.

Question 31

Sketch the graph of the parabola.

Question 32

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions.Parabola, directrix $y = 3$

Question 33

Complete the square to determine the type of curve represented by the equation.

A)
, parabola
B)
, ellipse
C)
, hyperbola
D)
, hyperbola
E) none of these

Question 34

Find an equation for the ellipse with center at the origin, major axis of length
, minor axis of length
, and whose foci lie on the
-axis.

Question 35

Find the lengths of the major and minor axes for the ellipse.

Question 36

Find an equation of the parabola whose graph is shown.

Question 37

Find the eccentricity and identify the conic given by $r= \frac { 2 } { 1 - \cos \theta }$
.

Question 38

Find the vertex, focus, and directrix of the parabola
.

Question 39

Find an equation of the parabola whose graph is shown.

Question 40

Find the foci of the hyperbola.

Question 41

Write a polar equation of a hyperbola with eccentricity $3$
and directrix $r = 2 \csc \theta$
.

Question 42

Find the vertices of the hyperbola.

A)
B)
C)
D)
E) none of these

Question 43

Determine the
- coordinates of
if the axes are rotated through an angle
.

Question 44

Sketch the graph of the parabola.

A)

B)

C)

D)

Question 45

Find the vertex, focus, and directrix of the parabola

A)
B)
C)
D)
E)

Question 46

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions.Parabola, directrix $y = 3$

A) $r = \frac { 3 } { 1 - \sin \theta }$

B) $r = \frac { 1 / 3 } { 1 + \sin \theta }$

C) $r= \frac { 3 } { 1 + \cos \theta }$

D) $r = \frac { 3 } { 1 - \cos \theta }$

E) none of these

Question 47

Find an equation for the conic whose graph is shown.

A)
B)
C)
D)
E)

Question 48

Find the foci of the hyperbola.

A)
B)
C)
D)
E)

Question 49

Find the vertex, and directrix of the parabola. $y ^ { 2 } + 2 y - 12 x + 37 = 0$

A) $V ( 6 , - 1 ) , x = 0$
B) $V ( 3 , - 1 )$ , $x = 0$
C) $V ( 6 , - 1 ) , y = 0$
D) $V ( 4 , - 1 ) , y = - 1$
E) $V ( 6,0 ) , x = - 1$

Question 50

A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 800 ft from the cannon and the highest point it reaches is 1600 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.

Question 51

Find an equation for the conic whose graph is shown.

A)
B)
C)
D)
E)

Question 52

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions.Parabola, directrix $y = - 3$

Question 53

Find an equation for the conic whose graph is shown.

A)
B)
C)
D)
E)

Question 54

Find an equation for the conic whose graph is shown.

A)
B)
C)
D)
E)

Question 55

Find the lengths of the major and minor axes for the ellipse.

A) major axis:
, minor:
B) major axis:
, minor:
C) major axis:
, minor:
D) major axis:
, minor:
E) none of these

Question 56

Find an equation of the parabola whose graph is shown.

A)
B)
C)
D)
E) none of these

Question 57

Find the vertices and foci for the ellipse. $y ^ { 2 } = 4 - 2 x ^ { 2 }$

A)Vertices: $( 0 , \pm \sqrt { 2 } )$ , foci: $( 0 , \pm 2 )$

B)Vertices: $( \pm 2,0 )$ , foci: $( 0 , \pm \sqrt { 2 } )$

C)Vertices: $( 0 , \pm 2 )$ , foci: $( 0 , \pm \sqrt { 2 } )$

D)Vertices: $( 0 , \pm 2 )$ , foci: $( \pm \sqrt { 2 } , 0 )$

E) none of these

Question 58

Find the foci for the hyperbola.

A)
B)
C)
D)
E) none

Question 59

Find an equation for the conic whose graph is shown.

A)
B)
C)
D)
E)

Question 60

Determine the
- coordinates of
if the axes are rotated through an angle
.

Question 61

Find the vertices of the hyperbola.

A)
B)
C)
D)
E) none of these

Question 62

Find the vertices and foci for the ellipse.

A)Vertices:
, foci:
B)Vertices:
, foci:
C)Vertices:
, foci:
D)Vertices:
, foci:
E) none of these

Question 63

Complete the square to determine the type of curve represented by the equation. $4 x ^ { 2 } + 9 y ^ { 2 } - 16 x - 20 = 0$

A) $\frac { ( x - 2 ) ^ { 2 } } { 9 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$ , ellipse

B) $9 ( x - 2 ) ^ { 2 } - 4 y ^ { 2 } = 9$ , hyperbola

C) $\frac { x ^ { 2 } } { 9 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$ , ellipse

D) $\frac { ( x - 2 ) ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ , hyperbola

E) $\frac { ( x - 2 ) ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ , ellipse

Question 64

Find the lengths of the major and minor axes for the ellipse.

A) major axis:
, minor:
B) major axis:
, minor:
C) major axis:
, minor:
D) major axis:
, minor:
E) none of these

Question 65

Find an equation for the parabola with vertex and directrix

A)
B)
C)
D)
E)

Question 66

Sketch the graph of the parabola.

A)

B)

C)

D)

Question 67

Find the eccentricity and identify the conic given by $r= \frac { 7 } { 1 + 3 \sin \theta }$

A) $e = 3$
, hyperbola
B) $e = \frac { 3 } { 7 }$
, ellipse
C) $e = \frac { 7 } { 3 }$
, hyperbola
D) $e = 7$
, hyperbola
E) none of these

Question 68

A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 ft from the cannon and the highest point it reaches is 2400 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.

A)
B)
C)
D)
E) none of these

Question 69

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola.

A)parabola
B)ellipse
C)hyperbola

Question 70

Sketch the graph of the ellipse.

A)

B)

C)
E)none of these
D)

Question 71

A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 800 ft from the cannon and the highest point it reaches is 1600 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.

Question 72

Find an equation for the hyperbola that has vertices and passes through the point

A)
B)
C)
D)
E) none of these

Question 73

Find an equation of a parabola with horizontal axis and vertex at the origin that passes through the point

A)
B)
C)
D)
E)

Question 74

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions.Parabola, directrix $y = - 3$

A) $r = \frac { 3 } { 1 - \sin \theta }$
B) $r= \frac { 3 } { 1 + \sin \theta }$
C) $r= \frac { 1 } { 1 - 3 \cos \theta }$
D) $r = \frac { 3 } { 1 + 3 \cos \theta }$
E) none of these

Question 75

Complete the square to determine the type of curve represented by the equation.

A)
, parabola
B)
, ellipse
C)
, hyperbola
D)
, parabola
E) none of these

Question 76

Write a polar equation of a hyperbola with eccentricity $3$ and directrix $r = 2 \csc \theta$

A) $r= \frac { 3 } { 1 + 3 \cos \theta }$

B) $r = \frac { 3 } { 1 - 3 \sin \theta }$

C) $r = \frac { 2 } { 2 + 3 \cos }$

D) $r = \frac { 6 } { 1 + 3 \sin \theta }$

E) $r= \frac { 6 } { 1 - 3 \sin \theta }$

Question 77

Find an equation for the ellipse with center at the origin, major axis of length $2 \sqrt { 7 }$ , minor axis of length $2 \sqrt { 3 }$
, and whose foci lie on the $y$
Axis

A) $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 7 } = 1$

B) $\frac { x ^ { 2 } } { 12 } + \frac { y ^ { 2 } } { 14 } = 1$

C) $\frac { x ^ { 2 } } { 14 } + \frac { y ^ { 2 } } { 12 } = 1$

D) $\frac { x ^ { 2 } } { 7 } + \frac { y ^ { 2 } } { 3 } = 1$

E) none of these

Question 78

Determine the $\overline { X Y }$ - coordinates of $( - 2,5 )$ If the axes are rotated through an angle $\phi = 30 ^ { \circ }$

A) $X = \frac { 5 - 2 \sqrt { 3 } } { 2 }$
, $Y = \frac { 2 + 5 \sqrt { 3 } } { 2 }$

B) $X = \frac { 2 \sqrt { 3 } } { 2 }$
, $Y = 1 - \frac { 5 \sqrt { 3 } } { 2 }$

C) $X = 5 - 2 \sqrt { 3 }$
, $Y = 1 + \frac { \sqrt { 3 } } { 2 }$

D) $X = \frac { 5 - 2 \sqrt { 3 } } { 2 }$
, $Y = \frac { \sqrt { 3 } + 5 } { 2 }$

E) $X = \frac { 1 + 2 \sqrt { 3 } } { 2 }$
, $Y = - 5 + \frac { \sqrt { 3 } } { 2 }$

Question 79

Find the vertex of the parabola.

A)
B)
C)
D)
E)

Question 80

Find an equation for the parabola with vertex and directrix
)