Question 1

Find the vertex, focus, directrix, and focal width of the parabola.
-$$\mathrm { y } ^ { 2 } = 32 \mathrm { x }$$

Accepted Answer

A)  Vertex: $( 0,0 )$; Focus: $( 8,0 )$; Directrix: $x = - 8$; Focal width: 128 
B)  Vertex: $( 0,0 )$; Focus: $( 0,8 )$; Directrix: $y = - 8$; Focal width: 8 
C)  Vertex: $( 0,0 )$; Focus: $( 8,0 )$; Directrix: $x = - 8$; Focal width: 32 
D)  Vertex: $( 0,0 )$; Focus: $( 8,8 )$; Directrix: $x = 8$; Focal width: 128 
A)  Vertex: $( 0,0 )$; Focus: $( 8,0 )$; Directrix: $x = - 8$; Focal width: 128 
B)  Vertex: $( 0,0 )$; Focus: $( 0,8 )$; Directrix: $y = - 8$; Focal width: 8 
C)  Vertex: $( 0,0 )$; Focus: $( 8,0 )$; Directrix: $x = - 8$; Focal width: 32 
D)  Vertex: $( 0,0 )$; Focus: $( 8,8 )$; Directrix: $x = 8$; Focal width: 128

Question 2

Solve the problem.
-A cross-section of an irrigation canal is a parabola. If the surface of the water is $40 \mathrm { ft }$ wide and the canal is $7 \mathrm { ft }$ deep at the center, how deep is it $10 \mathrm { ft }$ from the edge?

Accepted Answer

A)  $1.8 \mathrm { ft }$ 
B)  $3.1 \mathrm { ft }$ 
C)  $3.9 \mathrm { ft }$ 
D)  $5.2 \mathrm { ft }$ 
A)  $1.8 \mathrm { ft }$ 
B)  $3.1 \mathrm { ft }$ 
C)  $3.9 \mathrm { ft }$ 
D)  $5.2 \mathrm { ft }$

Question 3

Compute the distance between the points.
-$( a , b , c ) , ( - 4,8,6 )$

Accepted Answer

A)  $( ( a + 4 ) + ( b - 8 ) + ( c - 6 ) ) ^ { 2 }$ 
B)  $a + b + c - ( - 4 + 8 + 6 )$ 
C)  $\sqrt { ( a + 4 ) ^ { 2 } + ( b - 8 ) ^ { 2 } + ( c - 6 ) ^ { 2 } }$ 
D)  $( a + 4 ) ^ { 2 } + ( b - 8 ) ^ { 2 } + ( c - 6 ) ^ { 2 }$ 
A)  $( ( a + 4 ) + ( b - 8 ) + ( c - 6 ) ) ^ { 2 }$ 
B)  $a + b + c - ( - 4 + 8 + 6 )$ 
C)  $\sqrt { ( a + 4 ) ^ { 2 } + ( b - 8 ) ^ { 2 } + ( c - 6 ) ^ { 2 } }$ 
D)  $( a + 4 ) ^ { 2 } + ( b - 8 ) ^ { 2 } + ( c - 6 ) ^ { 2 }$

Question 4

Solve the problem.
-A satellite is to be put into an elliptical orbit around a moon. The moon is a sphere with radius of $149 \mathrm {~km}$. Determine an equation for the ellipse if the distance of the satellite from the surface of the moon varies from 910 $\mathrm { km }$ to $488 \mathrm {~km}$.

Accepted Answer

A)  $\frac { x ^ { 2 } } { 10592 } + \frac { y ^ { 2 } } { 637 ^ { 2 } } = 1$ 
B)  $\frac { x ^ { 2 } } { 488 } + \frac { y ^ { 2 } } { 910 } = 1$ 
C)  $\frac { x ^ { 2 } } { 637 ^ { 2 } } + \frac { y ^ { 2 } } { 10592 } = 1$ 
D)  $\frac { x ^ { 2 } } { 910 } + \frac { y ^ { 2 } } { 488 } = 1$ 
A)  $\frac { x ^ { 2 } } { 10592 } + \frac { y ^ { 2 } } { 637 ^ { 2 } } = 1$ 
B)  $\frac { x ^ { 2 } } { 488 } + \frac { y ^ { 2 } } { 910 } = 1$ 
C)  $\frac { x ^ { 2 } } { 637 ^ { 2 } } + \frac { y ^ { 2 } } { 10592 } = 1$ 
D)  $\frac { x ^ { 2 } } { 910 } + \frac { y ^ { 2 } } { 488 } = 1$

Question 5

Find the vertex, the focus, and the directrix of the parabola.
-$$2 x ^ { 2 } + 20 x - y + 53 = 0$$

Accepted Answer

A)  Vertex: $( - 5,0 )$; Focus: $( - 5,5 )$; Directrix: $x = 1$ 
B)  Vertex: $\left( - 5 , - \frac { 24 } { 17 } \right)$; Focus: $\left( - 5 , \frac { 23 } { 8 } \right) ;$ Directrix: $x = \frac { 25 } { 8 }$ 
C)  Vertex: $( - 5,3 )$; Focus: $( - 5,11 )$; Directrix: $x = 5$ 
D)  Vertex: $( - 5,3 )$; Focus: $\left( - 5 , \frac { 25 } { 8 } \right)$; Directrix: $y = \frac { 23 } { 8 }$ 
A)  Vertex: $( - 5,0 )$; Focus: $( - 5,5 )$; Directrix: $x = 1$ 
B)  Vertex: $\left( - 5 , - \frac { 24 } { 17 } \right)$; Focus: $\left( - 5 , \frac { 23 } { 8 } \right) ;$ Directrix: $x = \frac { 25 } { 8 }$ 
C)  Vertex: $( - 5,3 )$; Focus: $( - 5,11 )$; Directrix: $x = 5$ 
D)  Vertex: $( - 5,3 )$; Focus: $\left( - 5 , \frac { 25 } { 8 } \right)$; Directrix: $y = \frac { 23 } { 8 }$

Question 6

Find the standard form of the equation of the parabola.
-Vertex at the origin, opens downward, focal width $= 18$

Accepted Answer

A)  $x ^ { 2 } = 18 y$ 
B)  $x ^ { 2 } = - 18 y$ 
C)  $y ^ { 2 } = - 4.5 x$ 
D)  $y ^ { 2 } = 18 x$ 
A)  $x ^ { 2 } = 18 y$ 
B)  $x ^ { 2 } = - 18 y$ 
C)  $y ^ { 2 } = - 4.5 x$ 
D)  $y ^ { 2 } = 18 x$

Question 7

Match the given graph with its equation.
-

Accepted Answer

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

Find the vertex, focus, directrix, and focal width of the parabola. - $\mathrm { y } ^ { 2 } = 32 \mathrm { x }$

Solve the problem. -A cross-section of an irrigation canal is a parabola. If the surface of the water is $40 \mathrm { ft }$ wide and the canal is $7 \mathrm { ft }$ deep at the center, how deep is it $10 \mathrm { ft }$ from the edge?

Compute the distance between the points. - $( a , b , c ) , ( - 4,8,6 )$

Find the vertex, the focus, and the directrix of the parabola. - $2 x ^ { 2 } + 20 x - y + 53 = 0$

Find the standard form of the equation of the parabola. -Vertex at the origin, opens downward, focal width $= 18$

Match the given graph with its equation. -

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Exam 8: Analytic Geometry in Two and Three Dimensions

Find the vertex, focus, directrix, and focal width of the parabola. - y2=32x\mathrm { y } ^ { 2 } = 32 \mathrm { x }y2=32x

Solve the problem. -A cross-section of an irrigation canal is a parabola. If the surface of the water is 40ft40 \mathrm { ft }40ft wide and the canal is 7ft7 \mathrm { ft }7ft deep at the center, how deep is it 10ft10 \mathrm { ft }10ft from the edge?

Compute the distance between the points. - (a,b,c),(−4,8,6)( a , b , c ) , ( - 4,8,6 )(a,b,c),(−4,8,6)

Find the vertex, the focus, and the directrix of the parabola. - 2x2+20x−y+53=02 x ^ { 2 } + 20 x - y + 53 = 02x2+20x−y+53=0

Find the standard form of the equation of the parabola. -Vertex at the origin, opens downward, focal width =18= 18=18