Question 1

Match the given graph with its equation.
-

Accepted Answer

A)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
B)  $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 3 } = 1$ 
C)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$ 
D)  $\frac { x ^ { 2 } } { 10 } + \frac { y ^ { 2 } } { 6 } = 1$ 
A)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
B)  $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 3 } = 1$ 
C)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$ 
D)  $\frac { x ^ { 2 } } { 10 } + \frac { y ^ { 2 } } { 6 } = 1$

Question 2

Find the values of e, a, b, and c.
-$$r = \frac { 21 } { 5 - 2 \cos \theta }$$

Accepted Answer

A)  $\mathrm { e } = \frac { 2 } { 5 } , \mathrm { a } = 5 , \mathrm {~b} = \sqrt { 21 } , \mathrm { c } = 2$ 
B)  $\mathrm { e } = \frac { 2 } { 5 } , \mathrm { a } = 21 , \mathrm {~b} = 4 , \mathrm { c } = 5$ 
C)  $\mathrm { e } = \frac { 5 } { 2 } , \mathrm { a } = 2 , \mathrm {~b} = \sqrt { 21 } , \mathrm { c } = 5$ 
D)  $e = \frac { 5 } { 2 } , a = 21 , b = 4 , c = 5$ 
A)  $\mathrm { e } = \frac { 2 } { 5 } , \mathrm { a } = 5 , \mathrm {~b} = \sqrt { 21 } , \mathrm { c } = 2$ 
B)  $\mathrm { e } = \frac { 2 } { 5 } , \mathrm { a } = 21 , \mathrm {~b} = 4 , \mathrm { c } = 5$ 
C)  $\mathrm { e } = \frac { 5 } { 2 } , \mathrm { a } = 2 , \mathrm {~b} = \sqrt { 21 } , \mathrm { c } = 5$ 
D)  $e = \frac { 5 } { 2 } , a = 21 , b = 4 , c = 5$

Question 3

Find an equation in standard form for the hyperbola that satisfies the given conditions.
-Vertices at $( 0 , \pm 8 )$, asymptotes at $y = \pm \frac { 2 } { 5 } x$

Accepted Answer

A)  $\frac { y ^ { 2 } } { 64 } - \frac { x ^ { 2 } } { 100 } = 1$ 
B)  $\frac { y ^ { 2 } } { 64 } - \frac { x ^ { 2 } } { 400 } = 1$ 
C)  $\frac { y ^ { 2 } } { 400 } - \frac { x ^ { 2 } } { 64 } = 1$ 
D)  $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 16 } = 1$ 
A)  $\frac { y ^ { 2 } } { 64 } - \frac { x ^ { 2 } } { 100 } = 1$ 
B)  $\frac { y ^ { 2 } } { 64 } - \frac { x ^ { 2 } } { 400 } = 1$ 
C)  $\frac { y ^ { 2 } } { 400 } - \frac { x ^ { 2 } } { 64 } = 1$ 
D)  $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 16 } = 1$

Question 4

Match the given graph with its equation.
-

Accepted Answer

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

Question 5

Find the standard form of the equation of the parabola.
-Vertex at the origin, focus at $( 8,0 )$

Accepted Answer

A)  $y ^ { 2 } = 32 x$ 
B)  $y = \frac { 1 } { 32 } x ^ { 2 }$ 
C)  $x = \frac { 1 } { 32 } y ^ { 2 }$ 
D)  $x ^ { 2 } = 32 y$ 
A)  $y ^ { 2 } = 32 x$ 
B)  $y = \frac { 1 } { 32 } x ^ { 2 }$ 
C)  $x = \frac { 1 } { 32 } y ^ { 2 }$ 
D)  $x ^ { 2 } = 32 y$

Question 6

Find the linear equations that can be used to convert an (x, y) equation to a (u, v) equation using the given angle of
rotation
-$$\theta = \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$$

Accepted Answer

A)  $x = \frac { 4 } { 5 } u - \frac { 3 } { 5 } v$
$y = \frac { 3 } { 5 } u + \frac { 4 } { 5 } v$ 
B)  $\text {  } \begin{aligned}
x &=\frac{4}{3} u-\frac{4}{5} v \
y &=\frac{4}{5} u+\frac{4}{3} v
\end{aligned}$ 
C)  
$$
\begin{array}{l}
x=\frac{3}{5} u-\frac{4}{5} v \
y=\frac{4}{5} u+\frac{3}{5} v
\end{array}
$$ 
D)  $x = \frac { 5 } { 3 } u - \frac { 5 } { 4 } v$
 $y = \frac { 5 } { 4 } u + \frac { 5 } { 3 } v$ 
A)  $x = \frac { 4 } { 5 } u - \frac { 3 } { 5 } v$
$y = \frac { 3 } { 5 } u + \frac { 4 } { 5 } v$ 
B)  $\text {  } \begin{aligned}
x &=\frac{4}{3} u-\frac{4}{5} v \
y &=\frac{4}{5} u+\frac{4}{3} v
\end{aligned}$ 
C)  
$$
\begin{array}{l}
x=\frac{3}{5} u-\frac{4}{5} v \
y=\frac{4}{5} u+\frac{3}{5} v
\end{array}
$$ 
D)  $x = \frac { 5 } { 3 } u - \frac { 5 } { 4 } v$
 $y = \frac { 5 } { 4 } u + \frac { 5 } { 3 } v$

Question 7

Convert the equation to Cartesian form.
-$$r = \frac { - 66 } { 18 + 12 \sin \theta }$$

Accepted Answer

A)  $9 x ^ { 2 } + 5 y ^ { 2 } - 44 y - 121 = 0$ 
B)  $9 x ^ { 2 } + 5 y ^ { 2 } - 44 y + 121 = 0$ 
C)  $9 x ^ { 2 } + 13 y ^ { 2 } - 44 y - 121 = 0$ 
D)  $5 x ^ { 2 } + 44 x + 9 y ^ { 2 } + 121 = 0$ 
A)  $9 x ^ { 2 } + 5 y ^ { 2 } - 44 y - 121 = 0$ 
B)  $9 x ^ { 2 } + 5 y ^ { 2 } - 44 y + 121 = 0$ 
C)  $9 x ^ { 2 } + 13 y ^ { 2 } - 44 y - 121 = 0$ 
D)  $5 x ^ { 2 } + 44 x + 9 y ^ { 2 } + 121 = 0$

Question 8

Find the eccentricity of the hyperbola.
-$$x ^ { 2 } - y ^ { 2 } = 49$$

Accepted Answer

A)  0 
B)  $\sqrt { 7 }$ 
C)  $\sqrt { 2 }$ 
D)  1 
A)  0 
B)  $\sqrt { 7 }$ 
C)  $\sqrt { 2 }$ 
D)  1

Question 9

Match the polar equation with its graph.
-$$r = \frac { 10 } { 4 - 6 \sin \theta }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 10

Match the given graph with its equation.
-

Accepted Answer

A)  $\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y - 3 ) ^ { 2 } } { 4 } = 1$ 
B)  $\frac { ( y - 3 ) ^ { 2 } } { 16 } - \frac { ( x + 1 ) ^ { 2 } } { 4 } = 1$ 
C)  $\frac { ( y - 3 ) ^ { 2 } } { 4 } - \frac { ( x + 1 ) ^ { 2 } } { 16 } = 1$ 
D)  $\frac { ( x - 3 ) ^ { 2 } } { 16 } - \frac { ( x + 1 ) ^ { 2 } } { 4 } = 1$ 
A)  $\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y - 3 ) ^ { 2 } } { 4 } = 1$ 
B)  $\frac { ( y - 3 ) ^ { 2 } } { 16 } - \frac { ( x + 1 ) ^ { 2 } } { 4 } = 1$ 
C)  $\frac { ( y - 3 ) ^ { 2 } } { 4 } - \frac { ( x + 1 ) ^ { 2 } } { 16 } = 1$ 
D)  $\frac { ( x - 3 ) ^ { 2 } } { 16 } - \frac { ( x + 1 ) ^ { 2 } } { 4 } = 1$

Question 11

Find an equation in standard form for the ellipse that satisfies the given conditions.
-Minor axis endpoints $( \pm 4,0 )$, major axis length 22

Accepted Answer

A)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 16 } = 1$ 
B)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 11 } = 1$ 
C)  $\frac { x ^ { 2 } } { 11 } + \frac { y ^ { 2 } } { 4 } = 1$ 
D)  $\frac { y ^ { 2 } } { 121 } + \frac { x ^ { 2 } } { 16 } = 1$ 
A)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 16 } = 1$ 
B)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 11 } = 1$ 
C)  $\frac { x ^ { 2 } } { 11 } + \frac { y ^ { 2 } } { 4 } = 1$ 
D)  $\frac { y ^ { 2 } } { 121 } + \frac { x ^ { 2 } } { 16 } = 1$

Question 12

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.
-$$x ^ { 2 } + 4 x y + 6 y ^ { 2 } - 2 x + 4 y = 0$$

Accepted Answer

A)  Hyperbola 
B)  Parabola 
C)  Ellipse 
A)  Hyperbola 
B)  Parabola 
C)  Ellipse

Question 13

Use the given substitutions to convert the equation to a (u, v) equation with no cross-product term.
-$5 x ^ { 2 } - 6 x y + 5 y ^ { 2 } - 8 = 0 ;$ use $x = u / \sqrt { 2 } - v / \sqrt { 2 }$ and $y = u / \sqrt { 2 } + v / \sqrt { 2 }$

Accepted Answer

A)  $u ^ { 2 } + v ^ { 2 } = 4$ 
B)  $u ^ { 2 } + 4 v ^ { 2 } = 4$ 
C)  $u ^ { 2 } + 4 v ^ { 2 } = 1$ 
D)  $u ^ { 2 } + 8 v ^ { 2 } = 4$ 
A)  $u ^ { 2 } + v ^ { 2 } = 4$ 
B)  $u ^ { 2 } + 4 v ^ { 2 } = 4$ 
C)  $u ^ { 2 } + 4 v ^ { 2 } = 1$ 
D)  $u ^ { 2 } + 8 v ^ { 2 } = 4$

Question 14

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane.
-The hyperbola $x y = - 9$

Accepted Answer

A)  V: $( 3,3 ) , ( - 3 , - 3 )$
$F : ( 3 \sqrt { 2 } , 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , - 3 \sqrt { 2 } )$ 
B)  V: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$
F: $( 3 , - 3 ) , ( - 3,3 )$ 
C)  V: $( 3,3 ) , ( - 3 , - 3 )$
F: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$ 
D)  V: $( 3 , - 3 ) , ( - 3,3 )$
F: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$ 
A)  V: $( 3,3 ) , ( - 3 , - 3 )$
$F : ( 3 \sqrt { 2 } , 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , - 3 \sqrt { 2 } )$ 
B)  V: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$
F: $( 3 , - 3 ) , ( - 3,3 )$ 
C)  V: $( 3,3 ) , ( - 3 , - 3 )$
F: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$ 
D)  V: $( 3 , - 3 ) , ( - 3,3 )$
F: $( 3 \sqrt { 2 } , - 3 \sqrt { 2 } ) , ( - 3 \sqrt { 2 } , 3 \sqrt { 2 } )$

Question 15

Find the midpoint of the segment PQ.
-$$\mathrm { P } ( 4 , - 4 , - 8 ) , \mathrm { Q } ( 10 , - 1 , - 6 )$$

Accepted Answer

A)  $( 2 , - 2 , - 4 )$ 
B)  $( 7 , - 4 , - 6 )$ 
C)  $\left( 7 , - \frac { 5 } { 2 } , - 7 \right)$ 
D)  $\left( 5 , - \frac { 1 } { 2 } , - 3 \right)$ 
A)  $( 2 , - 2 , - 4 )$ 
B)  $( 7 , - 4 , - 6 )$ 
C)  $\left( 7 , - \frac { 5 } { 2 } , - 7 \right)$ 
D)  $\left( 5 , - \frac { 1 } { 2 } , - 3 \right)$

Question 16

Match the polar equation with its graph.
-$$r = \frac { 16 } { 8 - 7 \cos \theta }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 17

Solve the problem.
-An elliptical riding path is to be built on a rectangular piece of property that measures 8 mi by 4 mi. Find an equation for the ellipse if the path is to touch the center of the property line on all 4 sides

Accepted Answer

A)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 4 } = 1$ 
B)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 64 } = 1$ 
D)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$ 
A)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 4 } = 1$ 
B)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 64 } = 1$ 
D)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$

Question 18

Solve the problem.
- 
A tunnel is in the shape of a parabola. The maximum height is $30 \mathrm {~m}$ and it is $18 \mathrm {~m}$ wide at the base. What is the vertical clearance $2 \mathrm {~m}$ from the edge of the tunnel?

Accepted Answer

A)  $11.9 \mathrm {~m}$ 
B)  $28.5 \mathrm {~m}$ 
C)  $18.1 \mathrm {~m}$ 
D)  $1.5 \mathrm {~m}$ 
A)  $11.9 \mathrm {~m}$ 
B)  $28.5 \mathrm {~m}$ 
C)  $18.1 \mathrm {~m}$ 
D)  $1.5 \mathrm {~m}$

Question 19

Match the given graph with its equation.
-

Accepted Answer

A)  $25 x ^ { 2 } - 4 y ^ { 2 } = 100$ 
B)  $4 x ^ { 2 } - 25 y ^ { 2 } = 100$ 
C)  $4 y ^ { 2 } - 25 x ^ { 2 } = 100$ 
D)  $25 x ^ { 2 } + 4 y ^ { 2 } = 100$ 
A)  $25 x ^ { 2 } - 4 y ^ { 2 } = 100$ 
B)  $4 x ^ { 2 } - 25 y ^ { 2 } = 100$ 
C)  $4 y ^ { 2 } - 25 x ^ { 2 } = 100$ 
D)  $25 x ^ { 2 } + 4 y ^ { 2 } = 100$

Question 20

Graph the parabola.
-$(x+2)^{2}=4(y-5)$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Match the given graph with its equation. -

Find the values of e, a, b, and c. - $r = \frac { 21 } { 5 - 2 \cos \theta }$

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Vertices at $( 0 , \pm 8 )$ , asymptotes at $y = \pm \frac { 2 } { 5 } x$

Match the given graph with its equation. -

Find the standard form of the equation of the parabola. -Vertex at the origin, focus at $( 8,0 )$

Find the linear equations that can be used to convert an (x, y) equation to a (u, v) equation using the given angle of rotation - $\theta = \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$

Convert the equation to Cartesian form. - $r = \frac { - 66 } { 18 + 12 \sin \theta }$

Find the eccentricity of the hyperbola. - $x ^ { 2 } - y ^ { 2 } = 49$

Match the polar equation with its graph. - $r = \frac { 10 } { 4 - 6 \sin \theta }$

Match the given graph with its equation. -

Find an equation in standard form for the ellipse that satisfies the given conditions. -Minor axis endpoints $( \pm 4,0 )$ , major axis length 22

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $x ^ { 2 } + 4 x y + 6 y ^ { 2 } - 2 x + 4 y = 0$

Use the given substitutions to convert the equation to a (u, v) equation with no cross-product term. - $5 x ^ { 2 } - 6 x y + 5 y ^ { 2 } - 8 = 0 ;$ use $x = u / \sqrt { 2 } - v / \sqrt { 2 }$ and $y = u / \sqrt { 2 } + v / \sqrt { 2 }$

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane. -The hyperbola $x y = - 9$

Find the midpoint of the segment PQ. - $\mathrm { P } ( 4 , - 4 , - 8 ) , \mathrm { Q } ( 10 , - 1 , - 6 )$

Match the polar equation with its graph. - $r = \frac { 16 } { 8 - 7 \cos \theta }$

Solve the problem. -An elliptical riding path is to be built on a rectangular piece of property that measures 8 mi by 4 mi. Find an equation for the ellipse if the path is to touch the center of the property line on all 4 sides

Match the given graph with its equation. -

Graph the parabola. - $(x+2)^{2}=4(y-5)$ $Graph the parabola. - (x+2)^{2}=4(y-5)$

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Discrete Mathematics

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Exam 8: Analytic Geometry in Two and Three Dimensions

Match the given graph with its equation. -

Find the values of e, a, b, and c. - r=215−2cos⁡θr = \frac { 21 } { 5 - 2 \cos \theta }r=5−2cosθ21​

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Vertices at (0,±8)( 0 , \pm 8 )(0,±8) , asymptotes at y=±25xy = \pm \frac { 2 } { 5 } xy=±52​x

Match the given graph with its equation. -

Find the standard form of the equation of the parabola. -Vertex at the origin, focus at (8,0)( 8,0 )(8,0)

Find the linear equations that can be used to convert an (x, y) equation to a (u, v) equation using the given angle of rotation - θ=tan⁡−1(43)\theta = \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)θ=tan−1(34​)

Convert the equation to Cartesian form. - r=−6618+12sin⁡θr = \frac { - 66 } { 18 + 12 \sin \theta }r=18+12sinθ−66​

Find the eccentricity of the hyperbola. - x2−y2=49x ^ { 2 } - y ^ { 2 } = 49x2−y2=49

Match the polar equation with its graph. - r=104−6sin⁡θr = \frac { 10 } { 4 - 6 \sin \theta }r=4−6sinθ10​

Match the given graph with its equation. -

Find an equation in standard form for the ellipse that satisfies the given conditions. -Minor axis endpoints (±4,0)( \pm 4,0 )(±4,0) , major axis length 22

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - x2+4xy+6y2−2x+4y=0x ^ { 2 } + 4 x y + 6 y ^ { 2 } - 2 x + 4 y = 0x2+4xy+6y2−2x+4y=0

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane. -The hyperbola xy=−9x y = - 9xy=−9

Find the midpoint of the segment PQ. - P(4,−4,−8),Q(10,−1,−6)\mathrm { P } ( 4 , - 4 , - 8 ) , \mathrm { Q } ( 10 , - 1 , - 6 )P(4,−4,−8),Q(10,−1,−6)

Match the polar equation with its graph. - r=168−7cos⁡θr = \frac { 16 } { 8 - 7 \cos \theta }r=8−7cosθ16​

Solve the problem. -An elliptical riding path is to be built on a rectangular piece of property that measures 8 mi by 4 mi. Find an equation for the ellipse if the path is to touch the center of the property line on all 4 sides

Solve the problem. - A tunnel is in the shape of a parabola. The maximum height is 30 m30 \mathrm {~m}30 m and it is 18 m18 \mathrm {~m}18 m wide at the base. What is the vertical clearance 2 m2 \mathrm {~m}2 m from the edge of the tunnel?

Match the given graph with its equation. -

Graph the parabola. - (x+2)2=4(y−5)(x+2)^{2}=4(y-5)(x+2)2=4(y−5)

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Discrete Mathematics

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Find the values of e, a, b, and c. - $r = \frac { 21 } { 5 - 2 \cos \theta }$

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Vertices at $( 0 , \pm 8 )$ , asymptotes at $y = \pm \frac { 2 } { 5 } x$

Find the standard form of the equation of the parabola. -Vertex at the origin, focus at $( 8,0 )$

Find the linear equations that can be used to convert an (x, y) equation to a (u, v) equation using the given angle of rotation - $\theta = \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$

Convert the equation to Cartesian form. - $r = \frac { - 66 } { 18 + 12 \sin \theta }$

Find the eccentricity of the hyperbola. - $x ^ { 2 } - y ^ { 2 } = 49$

Match the polar equation with its graph. - $r = \frac { 10 } { 4 - 6 \sin \theta }$

Find an equation in standard form for the ellipse that satisfies the given conditions. -Minor axis endpoints $( \pm 4,0 )$ , major axis length 22

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $x ^ { 2 } + 4 x y + 6 y ^ { 2 } - 2 x + 4 y = 0$

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane. -The hyperbola $x y = - 9$

Find the midpoint of the segment PQ. - $\mathrm { P } ( 4 , - 4 , - 8 ) , \mathrm { Q } ( 10 , - 1 , - 6 )$

Match the polar equation with its graph. - $r = \frac { 16 } { 8 - 7 \cos \theta }$

Graph the parabola. - $(x+2)^{2}=4(y-5)$ $Graph the parabola. - (x+2)^{2}=4(y-5)$