Question 1

The Hyperbola
1 Locate a Hyperbola's Vertices and Foci
-$$\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 16 } = 1$$

Accepted Answer

A)  vertices: $( - 8,0 ) , ( 8,0 )$ 
B)  vertices: $( - 4,0 ) , ( 4,0 )$
foci: $( - 4 \sqrt { 5 } , 0 ) , ( 4 \sqrt { 5 } , 0 )$
foci: $( - 4 \sqrt { 5 } , 0 ) , ( 4 \sqrt { 5 } , 0 )$ 
C)  vertices: $( 0 , - 8 ) , ( 0,8 )$ 
D)  vertices: $( - 8,0 ) , ( 8,0 )$
foci: $( - 4 \sqrt { 5 } , 0 ) , ( 4 \sqrt { 5 } , 0 )$
foci: $( - 4,0 ) , ( 4,0 )$ 
A)  vertices: $( - 8,0 ) , ( 8,0 )$ 
B)  vertices: $( - 4,0 ) , ( 4,0 )$
foci: $( - 4 \sqrt { 5 } , 0 ) , ( 4 \sqrt { 5 } , 0 )$
foci: $( - 4 \sqrt { 5 } , 0 ) , ( 4 \sqrt { 5 } , 0 )$ 
C)  vertices: $( 0 , - 8 ) , ( 0,8 )$ 
D)  vertices: $( - 8,0 ) , ( 8,0 )$
foci: $( - 4 \sqrt { 5 } , 0 ) , ( 4 \sqrt { 5 } , 0 )$
foci: $( - 4,0 ) , ( 4,0 )$ A

Question 2

Eliminate the Parameter
-$$x = 6 \cos t , y = 6 \sin t ; 0 \leq t \leq 2 \pi$$

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } = 36 ; - 6 \leq x \leq 6$ 
B)  $x ^ { 2 } + y ^ { 2 } = 6 ; - 6 \leq x \leq 6$ 
C)  $x ^ { 2 } - y ^ { 2 } = 36 ; - 6 \leq x \leq 6$ 
D)  $x ^ { 2 } - y ^ { 2 } = 6 ; - 6 \leq x \leq 6$ 
A)  $x ^ { 2 } + y ^ { 2 } = 36 ; - 6 \leq x \leq 6$ 
B)  $x ^ { 2 } + y ^ { 2 } = 6 ; - 6 \leq x \leq 6$ 
C)  $x ^ { 2 } - y ^ { 2 } = 36 ; - 6 \leq x \leq 6$ 
D)  $x ^ { 2 } - y ^ { 2 } = 6 ; - 6 \leq x \leq 6$ A

Question 3

Find two sets of parametric equations for the given rectangular equation.
-$$y = 4 x + 6$$

Accepted Answer

A)  $x = t , y = 4 t + 6 ; x = \frac { t } { 4 } , y = t + 6$ 
B)  $x = t , y = 4 t + 6 ; x = 4 t , y = t + 6$ 
C)  $x = t , y = 4 t + 6 ; x = t , y = \frac { t } { 4 } + 6$ 
D)  $x = 4 t , y = t + 6 ; x = \frac { t } { 4 } , y = t + 6$ 
A)  $x = t , y = 4 t + 6 ; x = \frac { t } { 4 } , y = t + 6$ 
B)  $x = t , y = 4 t + 6 ; x = 4 t , y = t + 6$ 
C)  $x = t , y = 4 t + 6 ; x = t , y = \frac { t } { 4 } + 6$ 
D)  $x = 4 t , y = t + 6 ; x = \frac { t } { 4 } , y = t + 6$ A

Question 4

Solve Applied Problems Involving Ellipses
-The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the ar

Accepted Answer

A)  Truck 2 can pass under the bridge, but Truck 1 cannot. 
B)  Both Truck 1 and Truck 2 can pass under the bridge. 
C)  Neither Truck 1 nor Truck 2 can pass under the bridge. 
D)  Truck 1 can pass under the bridge, but Truck 2 cannot. 
A)  Truck 2 can pass under the bridge, but Truck 1 cannot. 
B)  Both Truck 1 and Truck 2 can pass under the bridge. 
C)  Neither Truck 1 nor Truck 2 can pass under the bridge. 
D)  Truck 1 can pass under the bridge, but Truck 2 cannot.

Question 5

Match the equation to the graph.
-$$x ^ { 2 } = - 7 y$$

Accepted Answer

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

Question 6

Graph the parabola.
-$$x ^ { 2 } = 12 y$$

Accepted Answer

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

Question 7

Graph the Polar Equations of Conics
-$\mathrm { r } = \frac { 4 } { 1 + 2 \cos \theta } \quad$ Identify the directrix and vertices.

Accepted Answer

A)  directrix: 2 unit(s) to the right of the pole at $x = - 2$
vertices: $( - 4 , \pi ) , \left( \frac { 4 } { 3 } , 0 \right)$
  
B)  directrix: 2 unit(s) to the left of the pole at $x = 2$ 
vertices: $( 4 , \pi ) , \left( - \frac { 4 } { 3 } , 0 \right)$
  
C)  directrix: 2 unit(s) above the pole at $\mathrm { y } = 2$
vertices: $\left( \frac { 4 } { 3 } , \frac { \pi } { 2 } \right) , \left( 4 , \frac { \pi } { 2 } \right)$
  
D)  directrix: 2 unit(s) below
the pole at $\mathrm { y } = - 2$
vertices: $\left( - \frac { 4 } { 3 } , \frac { \pi } { 2 } \right) , \left( - 4 , \frac { \pi } { 2 } \right)$
  
A)  directrix: 2 unit(s) to the right of the pole at $x = - 2$
vertices: $( - 4 , \pi ) , \left( \frac { 4 } { 3 } , 0 \right)$
  
B)  directrix: 2 unit(s) to the left of the pole at $x = 2$ 
vertices: $( 4 , \pi ) , \left( - \frac { 4 } { 3 } , 0 \right)$
  
C)  directrix: 2 unit(s) above the pole at $\mathrm { y } = 2$
vertices: $\left( \frac { 4 } { 3 } , \frac { \pi } { 2 } \right) , \left( 4 , \frac { \pi } { 2 } \right)$
  
D)  directrix: 2 unit(s) below
the pole at $\mathrm { y } = - 2$
vertices: $\left( - \frac { 4 } { 3 } , \frac { \pi } { 2 } \right) , \left( - 4 , \frac { \pi } { 2 } \right)$

Question 8

Identify Conics Without Rotating Axes
-$$10 x ^ { 2 } - 10 x y + 3 y ^ { 2 } - 3 x - 3 y + 4 = 0$$

Accepted Answer

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

Question 9

Use Rotation of Axes Formulas
-$7 x ^ { 2 } + 6 \sqrt { 3 } x y + y ^ { 2 } - 43 = 0 ; \theta = 30 ^ { \circ }$

Accepted Answer

A)  $\frac { x ^ { 2 } } { \frac { 43 } { 10 } } - \frac { y ^ { \prime 2 } } { \frac { 43 } { 2 } } = 1$ 
B)  $\frac { x ^ { \prime 2 } } { \frac { 43 } { 10 } } + \frac { y ^ { \prime 2 } } { \frac { 43 } { 2 } } = 1$ 
C)  $\frac { x ^ { \prime 2 } } { \frac { 1 } { 10 } } - \frac { y ^ { \prime 2 } } { \frac { 1 } { 2 } } = 1$ 
D)  $\frac { x ^ { \prime 2 } } { \frac { 1 } { 10 } } - \frac { y ^ { \prime 2 } } { \frac { 43 } { 2 } } = - 1$ 
A)  $\frac { x ^ { 2 } } { \frac { 43 } { 10 } } - \frac { y ^ { \prime 2 } } { \frac { 43 } { 2 } } = 1$ 
B)  $\frac { x ^ { \prime 2 } } { \frac { 43 } { 10 } } + \frac { y ^ { \prime 2 } } { \frac { 43 } { 2 } } = 1$ 
C)  $\frac { x ^ { \prime 2 } } { \frac { 1 } { 10 } } - \frac { y ^ { \prime 2 } } { \frac { 1 } { 2 } } = 1$ 
D)  $\frac { x ^ { \prime 2 } } { \frac { 1 } { 10 } } - \frac { y ^ { \prime 2 } } { \frac { 43 } { 2 } } = - 1$

Question 10

Graph the Polar Equations of Conics
-$r = \frac { 6 } { 2 + 2 \sin \theta } \quad$ Identify the directrix and vertex.

Accepted Answer

A)  directrix: 3 unit(s) above
the pole at $y = 3$ 
vertex: $\left( \frac { 3 } { 2 } , \frac { \pi } { 2 } \right)$
  
B)  directrix: 3 unit(s) to the right of
the pole at $x = 3$
vertex: $\left( \frac { 3 } { 2 } , 0 \right)$
  
C)  directrix: 3 unit(s) to the left of the pole at $x = - 3$
vertex: $\left( - \frac { 3 } { 2 } , 0 \right)$
  
D)  directrix: 3 unit(s) below
the pole at $\mathrm { y } = - 3$
vertex: $\left( - \frac { 3 } { 2 } , \frac { \pi } { 2 } \right)$
  
A)  directrix: 3 unit(s) above
the pole at $y = 3$ 
vertex: $\left( \frac { 3 } { 2 } , \frac { \pi } { 2 } \right)$
  
B)  directrix: 3 unit(s) to the right of
the pole at $x = 3$
vertex: $\left( \frac { 3 } { 2 } , 0 \right)$
  
C)  directrix: 3 unit(s) to the left of the pole at $x = - 3$
vertex: $\left( - \frac { 3 } { 2 } , 0 \right)$
  
D)  directrix: 3 unit(s) below
the pole at $\mathrm { y } = - 3$
vertex: $\left( - \frac { 3 } { 2 } , \frac { \pi } { 2 } \right)$

Question 11

Graph the ellipse and locate the foci.
-$$\frac { x ^ { 2 } } { 21 } + \frac { y ^ { 2 } } { 25 } = 1$$

Accepted Answer

A)  foci at $( 0,2 )$ and $( 0 , - 2 )$
  
B)  foci at $( 2,0 )$ and $( - 2,0 )$
  
C)  foci at $( 0 , \sqrt { 21 } )$ and $( 0 , - \sqrt { 21 } )$
  
D)  foci at $( 0,5 )$ and $( 0 , - 5 )$
  
A)  foci at $( 0,2 )$ and $( 0 , - 2 )$
  
B)  foci at $( 2,0 )$ and $( - 2,0 )$
  
C)  foci at $( 0 , \sqrt { 21 } )$ and $( 0 , - \sqrt { 21 } )$
  
D)  foci at $( 0,5 )$ and $( 0 , - 5 )$

Question 12

Write Equations of Parabolas in Standard Form
-Focus: $( - 5,4 )$; Directrix: $x = - 1$

Accepted Answer

A)  $( y - 4 ) ^ { 2 } = - 8 ( x + 3 )$ 
B)  $( x - 4 ) ^ { 2 } = - 8 ( y + 3 )$ 
C)  $( y + 3 ) ^ { 2 } = - 8 ( x - 4 )$ 
D)  $( x + 3 ) ^ { 2 } = - 8 ( y - 4 )$ 
A)  $( y - 4 ) ^ { 2 } = - 8 ( x + 3 )$ 
B)  $( x - 4 ) ^ { 2 } = - 8 ( y + 3 )$ 
C)  $( y + 3 ) ^ { 2 } = - 8 ( x - 4 )$ 
D)  $( x + 3 ) ^ { 2 } = - 8 ( y - 4 )$

Question 13

Identify Conics Without Rotating Axes
-$$3 x ^ { 2 } + 3 x y + 2 y ^ { 2 } + 2 x + 2 y - 7 = 0$$

Accepted Answer

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

Question 14

Additional Concepts
-$x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0$; Find the coordinates of the vertex.

Accepted Answer

A)  $( 0,0 )$ 
B)  $( - \sqrt { 2 } , 0 )$ 
C)  $( 2,0 )$ 
D)  $( - 2,0 )$ 
A)  $( 0,0 )$ 
B)  $( - \sqrt { 2 } , 0 )$ 
C)  $( 2,0 )$ 
D)  $( - 2,0 )$

Question 15

Solve Applied Problems Involving Parabolas
-An experimental model for a suspension bridge is built. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second tower. The towers are both 12.25 inches tall and stand 70 inches apart. At some point along the road from the lowest point of the cable, the cable is 1.1 inches above the roadway. Find the distance between that point and the base of the nearest tower.

Accepted Answer

A)  24.5 in. 
B)  10.3 in. 
C)  24.7 in. 
D)  10.7 in. 
A)  24.5 in. 
B)  10.3 in. 
C)  24.7 in. 
D)  10.7 in.

Question 16

Find a set of parametric equations for the rectangular equation.
-$$y = 4 x - 3$$

Accepted Answer

A)  $x = t ; y = 4 t - 3$ 
B)  $\mathrm { y } = 4 \mathrm { t } ; 4 \mathrm { x } = \mathrm { t } + 3$ 
C)  $x = \frac { t } { 4 } ; y = t - \frac { 3 } { 4 }$ 
D)  $x = t ; y = 4 t ^ { 2 } - 3$ 
A)  $x = t ; y = 4 t - 3$ 
B)  $\mathrm { y } = 4 \mathrm { t } ; 4 \mathrm { x } = \mathrm { t } + 3$ 
C)  $x = \frac { t } { 4 } ; y = t - \frac { 3 } { 4 }$ 
D)  $x = t ; y = 4 t ^ { 2 } - 3$

Question 17

Write Equations of Ellipses in Standard Form
-

Accepted Answer

A)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$
foci at $( - 4 \sqrt { 2 } , 0 )$ and $( 4 \sqrt { 2 } , 0 )$ 
B)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 36 } = 1$
 foci at $( - 4 \sqrt { 2 } , 0 )$ and $( 4 \sqrt { 2 } , 0 )$ 
C)  $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 4 } = 1$
foci at $( - 4 \sqrt { 2 } , 0 )$ and $( 4 \sqrt { 2 } , 0 )$ 
D)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$
foci at $( - 6,0 )$ and $( 6,0 )$ 
A)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$
foci at $( - 4 \sqrt { 2 } , 0 )$ and $( 4 \sqrt { 2 } , 0 )$ 
B)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 36 } = 1$
 foci at $( - 4 \sqrt { 2 } , 0 )$ and $( 4 \sqrt { 2 } , 0 )$ 
C)  $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 4 } = 1$
foci at $( - 4 \sqrt { 2 } , 0 )$ and $( 4 \sqrt { 2 } , 0 )$ 
D)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$
foci at $( - 6,0 )$ and $( 6,0 )$

Question 18

Match the equation to the graph.
-$$( y - 2 ) ^ { 2 } = 8 ( x - 2 )$$

Accepted Answer

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

Question 19

Additional Concepts
-$x ^ { 2 } + x y + y ^ { 2 } - 3 y - 6 = 0$; Find the coordinates of the vertices on the minor axis.

Accepted Answer

A)  $\left( \frac { \sqrt { 2 } } { 2 } , - \frac { 3 \sqrt { 2 } } { 2 } \right)$ and $\left( \frac { \sqrt { 2 } } { 2 } , \frac { 9 \sqrt { 2 } } { 2 } \right)$ 
B)  $\left( \frac { \sqrt { 2 } } { 2 } , \frac { 3 \sqrt { 2 } } { 2 } \right)$ and $\left( - \frac { \sqrt { 2 } } { 2 } , - \frac { 3 \sqrt { 2 } } { 2 } \right)$ 
C)  $( - \sqrt { 6 } , 0 )$ and $( \sqrt { 6 } , 0 )$ 
D)  $( 0 , - 2 )$ and $( 0,2 )$ 
A)  $\left( \frac { \sqrt { 2 } } { 2 } , - \frac { 3 \sqrt { 2 } } { 2 } \right)$ and $\left( \frac { \sqrt { 2 } } { 2 } , \frac { 9 \sqrt { 2 } } { 2 } \right)$ 
B)  $\left( \frac { \sqrt { 2 } } { 2 } , \frac { 3 \sqrt { 2 } } { 2 } \right)$ and $\left( - \frac { \sqrt { 2 } } { 2 } , - \frac { 3 \sqrt { 2 } } { 2 } \right)$ 
C)  $( - \sqrt { 6 } , 0 )$ and $( \sqrt { 6 } , 0 )$ 
D)  $( 0 , - 2 )$ and $( 0,2 )$

Question 20

Identify Conics Without Rotating Axes
-$$x ^ { 2 } - 3 x y - 3 y ^ { 2 } - 4 x + 2 y - 2 = 0$$

Accepted Answer

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

The Hyperbola 1 Locate a Hyperbola's Vertices and Foci - $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 16 } = 1$

Eliminate the Parameter - $x = 6 \cos t , y = 6 \sin t ; 0 \leq t \leq 2 \pi$

Find two sets of parametric equations for the given rectangular equation. - $y = 4 x + 6$

Solve Applied Problems Involving Ellipses -The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the ar

Match the equation to the graph. - $x ^ { 2 } = - 7 y$

Graph the parabola. - $x ^ { 2 } = 12 y$ $Graph the parabola. - x ^ { 2 } = 12 y$

Graph the Polar Equations of Conics - $\mathrm { r } = \frac { 4 } { 1 + 2 \cos \theta } \quad$ Identify the directrix and vertices. $Graph the Polar Equations of Conics - \mathrm { r } = \frac { 4 } { 1 + 2 \cos \theta } \quad Identify the directrix and vertices.$

Identify Conics Without Rotating Axes - $10 x ^ { 2 } - 10 x y + 3 y ^ { 2 } - 3 x - 3 y + 4 = 0$

Use Rotation of Axes Formulas - $7 x ^ { 2 } + 6 \sqrt { 3 } x y + y ^ { 2 } - 43 = 0 ; \theta = 30 ^ { \circ }$

Graph the Polar Equations of Conics - $r = \frac { 6 } { 2 + 2 \sin \theta } \quad$ Identify the directrix and vertex. $Graph the Polar Equations of Conics - r = \frac { 6 } { 2 + 2 \sin \theta } \quad Identify the directrix and vertex.$

Graph the ellipse and locate the foci. - $\frac { x ^ { 2 } } { 21 } + \frac { y ^ { 2 } } { 25 } = 1$ $Graph the ellipse and locate the foci. - \frac { x ^ { 2 } } { 21 } + \frac { y ^ { 2 } } { 25 } = 1$

Write Equations of Parabolas in Standard Form -Focus: $( - 5,4 )$ ; Directrix: $x = - 1$

Identify Conics Without Rotating Axes - $3 x ^ { 2 } + 3 x y + 2 y ^ { 2 } + 2 x + 2 y - 7 = 0$

Additional Concepts - $x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0$ ; Find the coordinates of the vertex.

Find a set of parametric equations for the rectangular equation. - $y = 4 x - 3$

Write Equations of Ellipses in Standard Form -

Match the equation to the graph. - $( y - 2 ) ^ { 2 } = 8 ( x - 2 )$

Additional Concepts - $x ^ { 2 } + x y + y ^ { 2 } - 3 y - 6 = 0$ ; Find the coordinates of the vertices on the minor axis.

Identify Conics Without Rotating Axes - $x ^ { 2 } - 3 x y - 3 y ^ { 2 } - 4 x + 2 y - 2 = 0$

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Matrices and Determinants

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

Filters

Exam 10: Conic Sections and Analytic Geometry

The Hyperbola 1 Locate a Hyperbola's Vertices and Foci - x264−y216=1\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 16 } = 164x2​−16y2​=1

Eliminate the Parameter - x=6cos⁡t,y=6sin⁡t;0≤t≤2πx = 6 \cos t , y = 6 \sin t ; 0 \leq t \leq 2 \pix=6cost,y=6sint;0≤t≤2π

Find two sets of parametric equations for the given rectangular equation. - y=4x+6y = 4 x + 6y=4x+6

Solve Applied Problems Involving Ellipses -The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the ar

Match the equation to the graph. - x2=−7yx ^ { 2 } = - 7 yx2=−7y

Graph the parabola. - x2=12yx ^ { 2 } = 12 yx2=12y

Graph the Polar Equations of Conics - r=41+2cos⁡θ\mathrm { r } = \frac { 4 } { 1 + 2 \cos \theta } \quadr=1+2cosθ4​ Identify the directrix and vertices.

Identify Conics Without Rotating Axes - 10x2−10xy+3y2−3x−3y+4=010 x ^ { 2 } - 10 x y + 3 y ^ { 2 } - 3 x - 3 y + 4 = 010x2−10xy+3y2−3x−3y+4=0

Use Rotation of Axes Formulas - 7x2+63xy+y2−43=0;θ=30∘7 x ^ { 2 } + 6 \sqrt { 3 } x y + y ^ { 2 } - 43 = 0 ; \theta = 30 ^ { \circ }7x2+63​xy+y2−43=0;θ=30∘

Graph the Polar Equations of Conics - r=62+2sin⁡θr = \frac { 6 } { 2 + 2 \sin \theta } \quadr=2+2sinθ6​ Identify the directrix and vertex.

Graph the ellipse and locate the foci. - x221+y225=1\frac { x ^ { 2 } } { 21 } + \frac { y ^ { 2 } } { 25 } = 121x2​+25y2​=1

Write Equations of Parabolas in Standard Form -Focus: (−5,4)( - 5,4 )(−5,4) ; Directrix: x=−1x = - 1x=−1

Identify Conics Without Rotating Axes - 3x2+3xy+2y2+2x+2y−7=03 x ^ { 2 } + 3 x y + 2 y ^ { 2 } + 2 x + 2 y - 7 = 03x2+3xy+2y2+2x+2y−7=0

Additional Concepts - x2+2xy+y2−8x+8y=0x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0x2+2xy+y2−8x+8y=0 ; Find the coordinates of the vertex.

Find a set of parametric equations for the rectangular equation. - y=4x−3y = 4 x - 3y=4x−3