Question 1

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

Accepted Answer

A)  $e = 3$
, hyperbola 
B)  $e = \frac { 3 } { 7 }$
, ellipse 
C)  $e = \frac { 7 } { 3 }$
, hyperbola 
D)  $e = 7$
, hyperbola 
E)  none of these 
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 2

Complete the square to determine the type of curve represented by the equation. $x ^ { 2 } + 4 x - 8 y + 36 = 0$

Accepted Answer

A)  $( x + 2 ) ^ { 2 } = 8 ( y - 5 )$
, parabola 
B)  $\frac { ( x + 2 ) ^ { 2 } } { 8 } + \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1$
, ellipse 
C)  $\frac { ( x + 2 ) ^ { 2 } } { 8 } - \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1$
, hyperbola 
D)  $( x + 2 ) ^ { 2 } = 4 ( y - 5 )$
, parabola 
E)  none of these 
A)  $( x + 2 ) ^ { 2 } = 8 ( y - 5 )$
, parabola 
B)  $\frac { ( x + 2 ) ^ { 2 } } { 8 } + \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1$
, ellipse 
C)  $\frac { ( x + 2 ) ^ { 2 } } { 8 } - \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1$
, hyperbola 
D)  $( x + 2 ) ^ { 2 } = 4 ( y - 5 )$
, parabola 
E)  none of these

Question 3

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

Accepted Answer

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 
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 4

Find an equation for the conic whose graph is shown.

Accepted Answer

A)  $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 25 } = 1$ 
B)  $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 16 } = 1$ 
C)  $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 16 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
E)  $9 y ^ { 2 } - \frac { x ^ { 2 } } { 16 } = 1$ 
A)  $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 25 } = 1$ 
B)  $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 16 } = 1$ 
C)  $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 16 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
E)  $9 y ^ { 2 } - \frac { x ^ { 2 } } { 16 } = 1$

Question 5

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.

Accepted Answer

A)  $( x - 1200 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 800 )$ 
B)  $( x - 800 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 2400 )$ 
C)  $( x - 800 ) ^ { 2 } = - \frac { 800 } { 3 } ( y - 2400 )$ 
D)  $( x - 2400 ) ^ { 2 } = - \frac { 800 } { 3 } ( y - 800 )$ 
E)  none of these 
A)  $( x - 1200 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 800 )$ 
B)  $( x - 800 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 2400 )$ 
C)  $( x - 800 ) ^ { 2 } = - \frac { 800 } { 3 } ( y - 2400 )$ 
D)  $( x - 2400 ) ^ { 2 } = - \frac { 800 } { 3 } ( y - 800 )$ 
E)  none of these

Question 6

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. $13 x ^ { 2 } - 10 x y + 13 y ^ { 2 } - 72 = 0$

Accepted Answer

A) parabola 
B) ellipse 
C) hyperbola 
A) parabola 
B) ellipse 
C) hyperbola

Question 7

Find the vertex, focus, and directrix of the parabola $x = \frac { 1 } { 8 } y ^ { 2 }$ 
.

Accepted Answer

Vertex @#LAT-DLM& 
,

Question 8

Find an equation of the parabola whose graph is shown.

Accepted Answer

The answer of Find an equation of the parabola whose...

Question 9

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

Accepted Answer

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 
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 10

Find an equation for the hyperbola that has vertices $( \pm 3,0 )$ and asymptotes $y = \pm 5 x$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 225 } - \frac { y ^ { 2 } } { 9 } = 1$ 
B)  $\frac { y ^ { 2 } } { 3 } - \frac { x ^ { 2 } } { 15 } = 1$ 
C)  $x ^ { 2 } - \frac { y ^ { 2 } } { 5 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 225 } = 1$ 
E)  none of these 
A)  $\frac { x ^ { 2 } } { 225 } - \frac { y ^ { 2 } } { 9 } = 1$ 
B)  $\frac { y ^ { 2 } } { 3 } - \frac { x ^ { 2 } } { 15 } = 1$ 
C)  $x ^ { 2 } - \frac { y ^ { 2 } } { 5 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 225 } = 1$ 
E)  none of these

Question 11

Find the foci of the hyperbola. $x ^ { 2 } - 5 y ^ { 2 } - 10 = 0$

Accepted Answer

The answer of Find the foci of the hyperbola. \(x...

Question 12

Find an equation for the parabola with vertex $( 5,5 )$ and directrix $y - a x i s$

Accepted Answer

A)  $( y - 5 ) ^ { 2 } = 25 x$ 
B)  $( y - 1 ) ^ { 2 } = 5 ( x - 5 )$ 
C)  $( x + 5 ) ^ { 2 } = 20 ( y + 5 )$ 
D)  $y ^ { 2 } = 20 x$ 
E)  $( y - 5 ) ^ { 2 } = 20 ( x - 5 )$ 
A)  $( y - 5 ) ^ { 2 } = 25 x$ 
B)  $( y - 1 ) ^ { 2 } = 5 ( x - 5 )$ 
C)  $( x + 5 ) ^ { 2 } = 20 ( y + 5 )$ 
D)  $y ^ { 2 } = 20 x$ 
E)  $( y - 5 ) ^ { 2 } = 20 ( x - 5 )$

Question 13

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

Accepted Answer

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 
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 14

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. $6 x ^ { 2 } + 3 y ^ { 2 } + x - 3 y = 1 - 3 \sqrt { 3 } x y$

Accepted Answer

A) parabola 
B) ellipse 
C) hyperbola 
A) parabola 
B) ellipse 
C) hyperbola

Question 15

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. $6 x ^ { 2 } + 3 y ^ { 2 } + x - 3 y = 1 - 3 \sqrt { 3 } x y$

Accepted Answer

The answer of Use the discriminant to determine whether the...

Question 16

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

Accepted Answer

The answer of Write a polar equation of a conic...

Question 17

Find an equation for the hyperbola that has vertices $( \pm \sqrt { 7 } , 0 )$ and passes through the point $( - 4,6 )$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 7 } = 1$ 
B)  $\frac { x ^ { 2 } } { 7 } - \frac { y ^ { 2 } } { 4 } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 28 } = 1$ 
D)  $\frac { x ^ { 2 } } { 7 } - \frac { y ^ { 2 } } { 28 } = 1$ 
E)  none of these 
A)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 7 } = 1$ 
B)  $\frac { x ^ { 2 } } { 7 } - \frac { y ^ { 2 } } { 4 } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 28 } = 1$ 
D)  $\frac { x ^ { 2 } } { 7 } - \frac { y ^ { 2 } } { 28 } = 1$ 
E)  none of these

Question 18

Find the vertices of the hyperbola. $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 36 } = 1$

Accepted Answer

A)  $V ( \pm 8 , \pm 6 )$ 
B)  $V ( \pm 6,0 )$ 
C)  $V ( \pm 8,0 )$ 
D)  $V ( 0 , \pm 8 )$ 
E)  none of these 
A)  $V ( \pm 8 , \pm 6 )$ 
B)  $V ( \pm 6,0 )$ 
C)  $V ( \pm 8,0 )$ 
D)  $V ( 0 , \pm 8 )$ 
E)  none of these

Question 19

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

Accepted Answer

The answer of Determine the $X Y$  - coordinates...

Question 20

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

Accepted Answer

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 
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

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

Complete the square to determine the type of curve represented by the equation. $x ^ { 2 } + 4 x - 8 y + 36 = 0$

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

Find an equation for the conic whose graph is shown.

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. $13 x ^ { 2 } - 10 x y + 13 y ^ { 2 } - 72 = 0$

Find the vertex, focus, and directrix of the parabola $x = \frac { 1 } { 8 } y ^ { 2 }$ .

Find an equation of the parabola whose graph is shown.

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

Find an equation for the hyperbola that has vertices $( \pm 3,0 )$ and asymptotes $y = \pm 5 x$

Find the foci of the hyperbola. $x ^ { 2 } - 5 y ^ { 2 } - 10 = 0$

Find an equation for the parabola with vertex $( 5,5 )$ and directrix $y - a x i s$

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

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. $6 x ^ { 2 } + 3 y ^ { 2 } + x - 3 y = 1 - 3 \sqrt { 3 } x y$

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. $6 x ^ { 2 } + 3 y ^ { 2 } + x - 3 y = 1 - 3 \sqrt { 3 } x y$

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

Find an equation for the hyperbola that has vertices $( \pm \sqrt { 7 } , 0 )$ and passes through the point $( - 4,6 )$

Find the vertices of the hyperbola. $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 36 } = 1$

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

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

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Matrices and Determinants

Sequences and Series

Counting and Probability

Filters

Exam 12: Conic Sections

Find the eccentricity and identify the conic given by r=71+3sin⁡θr= \frac { 7 } { 1 + 3 \sin \theta }r=1+3sinθ7​

Complete the square to determine the type of curve represented by the equation. x2+4x−8y+36=0x ^ { 2 } + 4 x - 8 y + 36 = 0x2+4x−8y+36=0

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

Find an equation for the conic whose graph is shown.

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. 13x2−10xy+13y2−72=013 x ^ { 2 } - 10 x y + 13 y ^ { 2 } - 72 = 013x2−10xy+13y2−72=0

Find the vertex, focus, and directrix of the parabola x=18y2x = \frac { 1 } { 8 } y ^ { 2 }x=81​y2 .

Find an equation of the parabola whose graph is shown.

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

Find an equation for the hyperbola that has vertices (±3,0)( \pm 3,0 )(±3,0) and asymptotes y=±5xy = \pm 5 xy=±5x

Find the foci of the hyperbola. x2−5y2−10=0x ^ { 2 } - 5 y ^ { 2 } - 10 = 0x2−5y2−10=0

Find an equation for the parabola with vertex (5,5)( 5,5 )(5,5) and directrix y−axisy - a x i sy−axis

Complete the square to determine the type of curve represented by the equation. 4x2+9y2−16x−20=04 x ^ { 2 } + 9 y ^ { 2 } - 16 x - 20 = 04x2+9y2−16x−20=0

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. 6x2+3y2+x−3y=1−33xy6 x ^ { 2 } + 3 y ^ { 2 } + x - 3 y = 1 - 3 \sqrt { 3 } x y6x2+3y2+x−3y=1−33​xy

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. 6x2+3y2+x−3y=1−33xy6 x ^ { 2 } + 3 y ^ { 2 } + x - 3 y = 1 - 3 \sqrt { 3 } x y6x2+3y2+x−3y=1−33​xy

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

Find an equation for the hyperbola that has vertices (±7,0)( \pm \sqrt { 7 } , 0 )(±7​,0) and passes through the point (−4,6)( - 4,6 )(−4,6)

Find the vertices of the hyperbola. x264−y236=1\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 36 } = 164x2​−36y2​=1

Determine the XYX YXY - coordinates of (6,0)( 6,0 )(6,0) if the axes are rotated through an angle ϕ=60∘\phi = 60 ^ { \circ }ϕ=60∘

Find an equation for the ellipse with center at the origin, major axis of length 272 \sqrt { 7 }27​ , minor axis of length 232 \sqrt { 3 }23​ , and whose foci lie on the yyy -axis

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Matrices and Determinants

Sequences and Series

Counting and Probability

Filters

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

Complete the square to determine the type of curve represented by the equation. $x ^ { 2 } + 4 x - 8 y + 36 = 0$

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

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. $13 x ^ { 2 } - 10 x y + 13 y ^ { 2 } - 72 = 0$

Find the vertex, focus, and directrix of the parabola $x = \frac { 1 } { 8 } y ^ { 2 }$ .

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

Find an equation for the hyperbola that has vertices $( \pm 3,0 )$ and asymptotes $y = \pm 5 x$

Find the foci of the hyperbola. $x ^ { 2 } - 5 y ^ { 2 } - 10 = 0$

Find an equation for the parabola with vertex $( 5,5 )$ and directrix $y - a x i s$

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

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. $6 x ^ { 2 } + 3 y ^ { 2 } + x - 3 y = 1 - 3 \sqrt { 3 } x y$

Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or a hyperbola. $6 x ^ { 2 } + 3 y ^ { 2 } + x - 3 y = 1 - 3 \sqrt { 3 } x y$

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

Find an equation for the hyperbola that has vertices $( \pm \sqrt { 7 } , 0 )$ and passes through the point $( - 4,6 )$

Find the vertices of the hyperbola. $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 36 } = 1$

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

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