Deck 8: Conic Sections and Quadratic Systems

Question 1

Find the equation of the hyperbola with center $( 3 , - 3 ) , a ^ { 2 } = 49$ and $b ^ { 2 } = 25$ .
a. $\quad \frac { ( x - 3 ) ^ { 2 } } { 49 } - \frac { ( y - 3 ) ^ { 2 } } { 25 } = 1$
b. $\quad \frac { ( x - 3 ) ^ { 2 } } { 49 } - \frac { ( y + 3 ) ^ { 2 } } { 25 } = 1$
c. $\quad \frac { ( x + 3 ) ^ { 2 } } { 49 } - \frac { ( y + 3 ) ^ { 2 } } { 25 } = 1$
d. $\quad \frac { ( x - 3 ) ^ { 2 } } { 25 } - \frac { ( y + 3 ) ^ { 2 } } { 49 } = 1$
e. $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 25 } = 1$

Answer 1

B

Question 2

Parallel beams of similarly charged particles are shot from two atomic accelerators $L = 24$ meters apart, as shown in the figure below. If the particles were not deflected, the beams would be $H = 2.4 \times 10 ^ { - 4 }$ meters apart. However, because the charged particles repel each other, the beams follow the hyperbolic path $y = \frac { k } { x }$ , for some $k$ . Find $k$ .


a. $\quad k = 2.88 \times 10 ^ { - 3 }$
b. $\quad k = 1.44 \times 10 ^ { - 3 }$
c. $\quad k = 1.44$
d. $\quad k = 1.44 \times 10 ^ { - 6 }$
e. none of these

Answer 2

B

Question 3

An arch is a semi-ellipse 16 meters wide and 8 meters high.Write the equation of the ellipse if the ellipse is centered at the origin. a. $\quad \frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 64 } = 1$
b. $\frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 16 } = 1$
c. $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 64 } = 1$
d. $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 8 } = 1$
e. none of these

Answer 3

A

Question 4

Find the equation of the parabola with vertex at $( 5,6 )$ and which passes through $( 4,8 )$ and $( 4,4 )$ .
a. $\quad ( y - 6 ) ^ { 2 } = - 8 ( x - 5 )$
b. $( y - 4 ) ^ { 2 } = 2 ( x - 4 )$
c. $( y - 6 ) ^ { 2 } = - 4 ( x + 5 )$
d. $( y + 6 ) ^ { 2 } = 4 ( x + 5 )$
e. $( y - 6 ) ^ { 2 } = - 4 ( x - 5 )$

Question 5

Find the equation of the curve on which the point $P$ lies. The distance between point $P$ and the point ( 0 ,
11) is $\frac { 11 } { 7 }$ of the distance between point $P$ and the line $y = - 7$ .
a. $\quad 49 x ^ { 2 } - 72 y ^ { 2 } + 2,772 y = 0$
b. $\quad 2,772 x ^ { 2 } - 72 y ^ { 2 } - 49 y = 0$
c. $\quad 49 x ^ { 2 } - 72 y ^ { 2 } - 2,772 y = 0$
d. $\quad 49 x ^ { 2 } - 72 y ^ { 2 } - 2,772 y = 50$
e. none of these

Question 6

Find the equation of the circle with a radius of 9 and its center at the intersection of $5 x + y = 20$ and $2 x -$ $3 y = - 9$ .
a. $\quad ( x - 3 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 9$
b. $\quad ( x - 3 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 81$
c. $\quad ( x + 3 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 81$
d. $\quad ( x + 3 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 9$
e. $\quad ( x - 3 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 81$

Question 7

Find the equation of the hyperbola with center $( 1,3 ) , a = 2$ and $b = 4$ . Transverse axis is horizontal.
a.
$$\frac { ( x - 1 ) ^ { 2 } } { 4 } - \frac { ( y - 3 ) ^ { 2 } } { 16 } = 1$$
b.
$$\frac { ( x - 2 ) ^ { 2 } } { 2 } - \frac { ( y - 3 ) ^ { 2 } } { 16 } = 1$$
c.
$$\frac { ( x + 1 ) ^ { 2 } } { 4 } - \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$$
d.
$$\frac { ( x - 1 ) ^ { 2 } } { 4 } + \frac { ( y - 3 ) ^ { 2 } } { 16 } = 1$$

Question 8

Write the equation of the ellipse that has its center at the origin, focus at $( 2,0 )$ and $\frac { 3 } { 2 }$ is one-half the length of the minor axis.
a. $\quad \frac { 4 x ^ { 2 } } { 25 } + \frac { 4 y ^ { 2 } } { 9 } = 1$
b. $\quad \frac { 4 x ^ { 2 } } { 25 } + \frac { 4 y ^ { 2 } } { 9 } = - 1$
c. $\quad \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$
d. $\quad \frac { 2 x ^ { 2 } } { 3 } + \frac { 2 y ^ { 2 } } { 5 } = 1$
e. $\quad \frac { 4 x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$

Question 9

Find the graph of the following equation. $$\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { 9 y ^ { 2 } } { 25 } = 1$$
a.


b.

c.

Question 10

Find the equation of the parabola with vertex at $( 0,0 )$ and focus at $( 0,4 )$ .
a. $\quad x ^ { 2 } = 16 y$
b. $\quad y ^ { 2 } = 16 x$
c. $\quad x ^ { 2 } = 4 y$
d. $x ^ { 2 } = y + 4$
e. none of these

Question 11

Write the equation of the ellipse with center at $( 5,1 ) , a = 5 , b = 2$ and major axis parallel to the x-axis.
a. $\quad \frac { ( x - 5 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
b. $\quad \frac { ( x - 5 ) ^ { 2 } } { 4 } + \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$
c. $\quad \frac { ( x - 5 ) ^ { 2 } } { 25 } - \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
d.
$$\frac { ( x - 5 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = - 1$$

Question 12

Identify the conic as a circle, parabola, ellipse, or hyperbola. $$- 6 x ^ { 2 } = - 48 x + 11 \left( y ^ { 2 } - 10 y \right) + 305$$
a. parabola
b. hyperbola
c. ellipse
d. circle
e. none of the above

Question 13

Find the equation of the ellipse with center at $( 8,9 )$ , vertex at $( 16,9 )$ , and passing through the intersection of $y = - 2 x + 27$ and $y = x + 3$ .
a. $\frac { ( x - 8 ) ^ { 2 } } { 8 } + \frac { ( y - 9 ) ^ { 2 } } { 2 } = 1$
b. $\frac { ( x - 8 ) ^ { 2 } } { 49 } + \frac { ( y - 9 ) ^ { 2 } } { 9 } = 1$
c. $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 9 ) ^ { 2 } } { 4 } = 1$
d. $\frac { ( x - 8 ) ^ { 2 } } { 49 } + \frac { ( y - 9 ) ^ { 2 } } { 4 } = 1$
e. $\frac { ( x - 8 ) ^ { 2 } } { 81 } + \frac { ( y - 9 ) ^ { 2 } } { 9 } = 1$

Question 14

Find the equation of the parabola passing through the given points. $( - 2 , - 5 ) , ( 1,7 )$ , and $( 2,15 )$
a. $\quad y = x ^ { 2 } - 5 x + 1$
b. $\quad y = x ^ { 2 } + 5 x + 1$
c. $\quad y = 5 x + 7$
d. $\quad y = - 3 x ^ { 2 } + 5 x + 17$
e. none of these

Question 15

Find the equation of the circle graphed below.
a. $\quad x ^ { 2 } + y ^ { 2 } = 1$
b. $\quad x ^ { 2 } + y ^ { 2 } = 25$
c. $x ^ { 2 } + y ^ { 2 } = 5$
d. $x ^ { 2 } + y = 25$
e. $y ^ { 2 } = x ^ { 2 } + 25$

Question 16

Find the equation of the circle with a radius of 2 and its center at the intersection of $2 x + y = 2$ and $- 4 x -$ $5 y = 2$ .
a. $\quad ( x + 2 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 2$
b. $( x + 2 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 4$
c. $\quad ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 2$
d. $\quad ( x - 2 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 4$
e. none of these

Question 17

Stones dropped into a calm pond at points $A$ and $B$ create ripples that propagate in widening circles. In the figure below, points $A$ and $B$ are 26 feet apart, and the radii of the circles differ by 10 feet. The point $P ( x , y )$ where the circles intersect moves along a curve. Find the equation of the curve.


a.

$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 144 } = 1$$
b.
$$\frac { x ^ { 2 } } { 27 } + \frac { y ^ { 2 } } { 288 } = 1$$
c. $\quad \frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 144 } = 1$
d.
$$\frac { ( x - 10 ) ^ { 2 } } { 25 } - \frac { ( y - 1 ) ^ { 2 } } { 144 } = 1$$

Question 18

Find the area of the fundamental rectangle of the hyperbola. $$9 ( x - 5 ) ^ { 2 } - 81 ( y - 3 ) ^ { 2 } = 729$$
a. $\quad 27$ square units
b. $\quad 54$ square units
c. $\quad 216$ square units
d. 108 square units
e. none of these

Question 19

Graph the following equation. $$x ^ { 2 } - 6 x + y ^ { 2 } = 7$$
a.


b.

c.

Question 20

Tell whether the parabolic graph of the equation opens up, down, to the left, or to the right. $$y ^ { 2 } = - 4 x$$
a. opens down
b. opens up
c. opens to the right
d. opens to the left

Question 21

Graph the following equation. $$x ^ { 2 } - 8 x + y ^ { 2 } = 0$$
a.

b.

c.

Question 22

Tell whether the parabolic graph of the equation opens up, down, to the left, or to the right. $$y ^ { 2 } = - 8 x$$
a. opens down
b. opens up
c. opens to the right
d. opens to the left

Question 23

Find the equation of the hyperbola with center $( 2 , - 2 ) , a ^ { 2 } = 25$ and $b ^ { 2 } = 4$ .
a. $\quad \frac { ( x - 2 ) ^ { 2 } } { 25 } - \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$
b. $\quad \frac { ( x - 2 ) ^ { 2 } } { 4 } - \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$
c. $\quad \frac { ( x - 2 ) ^ { 2 } } { 25 } - \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$
d. $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1$
e. $\quad \frac { ( x + 2 ) ^ { 2 } } { 25 } - \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$

Question 24

Find the equation of the hyperbola with center $( 3,1 ) , a = 5$ and $b = 4$ . Transverse axis is horizontal.
a. $\frac { ( x - 3 ) ^ { 2 } } { 25 } - \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1$
b. $\quad \frac { ( x - 3 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1$
c. $\quad \frac { ( x - 4 ) ^ { 2 } } { 23 } - \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1$
d. $\quad \frac { ( x + 3 ) ^ { 2 } } { 25 } - \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$
e. none of these

Question 25

Write the equation of the ellipse that has its center at the origin, focus at $( 1,0 )$ and $\frac { 3 } { 4 }$ is one-half the length of the minor axis.
a. $\quad \frac { 16 x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$
b. $\quad \frac { 16 x ^ { 2 } } { 25 } + \frac { 16 y ^ { 2 } } { 9 } = - 1$
c. $\quad \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$
d. $\quad \frac { 4 x ^ { 2 } } { 3 } + \frac { 4 y ^ { 2 } } { 7 } = 1$
e. $\quad \frac { 16 x ^ { 2 } } { 25 } + \frac { 16 y ^ { 2 } } { 9 } = 1$

Question 26

Find the equation of the ellipse with center at $( 7,7 )$ , vertex at $( 16,7 )$ , and passing through the intersection of $y = - 2 x + 25$ and $y = x + 4$ .
a.
$$\frac { ( x - 7 ) ^ { 2 } } { 9 } + \frac { ( y - 7 ) ^ { 2 } } { 4 } = 1$$
b.

$$\frac { ( x - 7 ) ^ { 2 } } { 64 } + \frac { ( y - 7 ) ^ { 2 } } { 16 } = 1$$
c.
$$\frac { ( x - 7 ) ^ { 2 } } { 100 } + \frac { ( y - 7 ) ^ { 2 } } { 25 } = 1$$
d.

$$\frac { ( x - 7 ) ^ { 2 } } { 81 } + \frac { ( y - 7 ) ^ { 2 } } { 16 } = 1$$
e.
$$\frac { ( x - 7 ) ^ { 2 } } { 64 } + \frac { ( y - 7 ) ^ { 2 } } { 25 } = 1$$

Question 27

Find the equation of the circle with a radius of 4 and its center at the intersection of $4 x + y = 2$ and $- 3 x -$ $4 y = 5$ .
a. $\quad ( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$
b. $\quad ( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$
c. $( x - 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$
d. $( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 4$
e. none of these

Question 28

Find the equation of the parabola with vertex at (0, 0) and focus at (0, 5). a. $\quad x ^ { 2 } = y + 5$
b. $\quad y ^ { 2 } = 20 x$
c. $\quad x ^ { 2 } = 5 y$
d. $\quad x ^ { 2 } = 20 y$
e. none of these

Question 29

Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 73 \\y = x ^ { 2 } - 61\end{array} \right.$$
a. $( 3,8 ) , ( - 8,3 )$
b. $\quad ( 5,3 ) , ( 3 , - 8 )$
c. $\quad ( 8,3 ) , ( 3 , - 8 )$
d. $\quad ( 8,3 ) , ( - 8,3 )$
e. $\quad ( - 8,3 ) , ( 8,5 )$

Question 30

An arch is a semi-ellipse 16 meters wide and 8 meters high. Write the equation of the ellipse if the ellipse is centered at the origin.
a. $\quad \frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 8 } = 1$
b. $\quad \frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 64 } = 1$
c. $\quad \frac { x ^ { 2 } } { 8 } + \frac { y ^ { 2 } } { 16 } = 1$
d. $\quad \frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 64 } = 1$
e. none of these

Question 31

Find the equation of the circle graphed below.
a. $\quad y ^ { 2 } = x ^ { 2 } + 25$
b. $\quad x ^ { 2 } + y ^ { 2 } = 1$
c. $\quad x ^ { 2 } + y ^ { 2 } = 25$
d. $\quad x ^ { 2 } + y ^ { 2 } = 5$
e. $\quad x ^ { 2 } + y = 25$

Question 32

Write the equation of the ellipse with center at $( 2,3 ) , a = 5 , b = 3$ and major axis parallel to the x-axis.
a.
$$\frac { ( x - 2 ) ^ { 2 } } { 25 } + \frac { ( y - 3 ) ^ { 2 } } { 9 } = 1$$
b.
$$\frac { ( x - 2 ) ^ { 2 } } { 25 } + \frac { ( y - 3 ) ^ { 2 } } { 9 } = - 1$$
c.
$$\frac { ( x - 2 ) ^ { 2 } } { 25 } - \frac { ( y - 3 ) ^ { 2 } } { 9 } = 1$$
d.
$$\frac { ( x - 2 ) ^ { 2 } } { 9 } + \frac { ( y - 3 ) ^ { 2 } } { 25 } = 1$$

Question 33

Find the graph of the following equation. $$\frac { ( x - 4 ) ^ { 2 } } { 25 } + \frac { 16 y ^ { 2 } } { 49 } = 1$$
a.

b.

c.

Question 34

Find the equation of the parabola with vertex at (8, 3) and which passes through (7, 5) and (7, 1). a. $\quad ( y - 3 ) ^ { 2 } = - 4 ( x + 8 )$
b. $( y - 3 ) ^ { 2 } = - 4 ( x - 8 )$
c. $( y - 1 ) ^ { 2 } = 2 ( x - 7 )$
d. $( y - 3 ) ^ { 2 } = - 8 ( x - 8 )$
e. $( y + 3 ) ^ { 2 } = 4 ( x + 8 )$

Question 35

Jim drove 252 miles. Jim's brother made the same trip at a speed $21.6$ miles per hour slower than Jim did and required an extra $1 \frac { 1 } { 2 }$ hours. Find Jim's rate and time.
a. $\quad 72 \mathrm { mph } , 2.5 \mathrm { hr }$
b. $\quad 66 \mathrm { mph } , 3.5 \mathrm { hr }$
c. $\quad 72 \mathrm { mph } , 3.5 \mathrm { hr }$
d. $\quad 66 \mathrm { mph } , 2.5 \mathrm { hr }$
e. not possible

Question 36

Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 97 \\x + y = 13\end{array} \right.$$
a. $\quad ( 4,9 ) , ( 9,4 )$
b. $\quad ( 1,9 ) , ( 9,1 )$
c. $\quad ( 9,4 ) , ( 9,4 )$
d. $( 4,9 ) , ( 4,1 )$
e. no solution

Question 37

Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l } x ^ { 2 } - 8 x - y = - 15 \\x ^ { 2 } - 8 x + y = - 15\end{array} \right.$$
a. $\quad ( 4,0 ) , ( 5,0 )$
b. $\quad ( 0,4 ) , ( 3,0 )$
c. $\quad ( 0,4 ) , ( 0,5 )$
d. $( 3,0 ) , ( 5,0 )$
e. no solution

Question 38

Find the equation of the circle with a radius of 9 and its center at the intersection of $3 x + y = 5$ and $2 x -$ $3 y = - 4$ .
a. $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 9$
b. $\quad ( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 81$
c. $( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 9$
d. $\quad ( x - 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 81$
e. $\quad ( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 81$

Question 39

The area of a rectangle is 30 square centimeters, and its perimeter is 22 centimeters.Find the dimensions of the rectangle.

A)22 cm by 22 cm
B)5 cm by 5 cm
C)5 cm by 6 cm
D)5 cm by 3 cm
E)3 cm by 6 cm

Question 40

Find the equation of the parabola passing through the given points. $( - 2,6 ) , ( 1,12 )$ , and $( 2,18 )$
a. $\quad y = - 2 x ^ { 2 } + 3 x + 20$
b. $y = x ^ { 2 } + 3 x + 8$
c. $y = 3 x + 3$
d. $\quad y = x ^ { 2 } - 3 x + 8$
e. none of these

Question 41

Parallel beams of similarly charged particles are shot from two atomic accelerators $L = 28$ meters apart, as shown in the figure below. If the particles were not deflected, the beams would be $H = 2.6 \times 10 ^ { - 4 }$ meters apart. However, because the charged particles repel each other, the beams follow the hyperbolic path $y = \frac { k } { x }$ , for some $k$ . Find $k$ .


a. $\quad k = 1.82 \times 10 ^ { - 6 }$
b. $k = 3.64 \times 10 ^ { - 3 }$
c. $k = 1.82 \times 10 ^ { - 3 }$
d. $k = 1.82$
e. none of these

Question 42

Jim drove 196 miles. Jim's brother made the same trip at a speed $16.8$ miles per hour slower than Jim did and required an extra $1 \frac { 1 } { 2 }$ hours. Find Jim's rate and time.
a. $\quad 56 \mathrm { mph } , 4.5 \mathrm { hr }$
b. $\quad 66 \mathrm { mph } , 3.5 \mathrm { hr }$
c. $\quad 66 \mathrm { mph } , 4.5 \mathrm { hr }$
d. $\quad 56 \mathrm { mph } , 3.5 \mathrm { hr }$
e. not possible

Question 43

Identify the conic as a circle, parabola, ellipse, or hyperbola. $$2 x ^ { 2 } = 8 x + 3 \left( y ^ { 2 } - 8 y \right) + 46$$
a. circle
b. hyperbola
c. parabola
d. ellipse
e. none of the above

Question 44

Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 41 \\y = x ^ { 2 } - 11\end{array} \right.$$
a. $\quad ( 4,5 ) , ( - 4,5 )$
b. $\quad ( 5,4 ) , ( - 4,5 )$
c. $\quad ( 4,5 ) , ( 5 , - 4 )$
d. $\quad ( - 4,5 ) , ( 4,9 )$
e. $\quad ( 9,5 ) , ( 5 , - 4 )$

Question 45

Graph the following equation. $$x ^ { 2 } - 8 x + y ^ { 2 } = 0$$
a.


b.

c.

Question 46

Find the area of the fundamental rectangle of the hyperbola. $$16 ( x - 4 ) ^ { 2 } - 81 ( y - 2 ) ^ { 2 } = 1,296$$
a. 288 square units
b. 144 square units
c. $\quad 72$ square units
d. 36 square units
e. none of these

Question 47

Find the equation of the parabola with vertex at $( 7,5 )$ and which passes through $( 6,7 )$ and $( 6,3 )$ .
a. $( y - 5 ) ^ { 2 } = - 8 ( x - 7 )$
b. $\quad ( y - 3 ) ^ { 2 } = 2 ( x - 6 )$
c. $( y - 5 ) ^ { 2 } = - 4 ( x - 7 )$
d. $( y + 5 ) ^ { 2 } = 4 ( x + 7 )$
e. $( y - 5 ) ^ { 2 } = - 4 ( x + 7 )$

Question 48

Write the equation of the ellipse with center at $( 1,4 ) , a = 3 , b = 2$ and major axis parallel to the $x$ -axis.
a. $\quad \frac { ( x - 1 ) ^ { 2 } } { 9 } + \frac { ( y - 4 ) ^ { 2 } } { 4 } = 1$
b. $\quad \frac { ( x - 1 ) ^ { 2 } } { 9 } - \frac { ( y - 4 ) ^ { 2 } } { 4 } = 1$
c. $\quad \frac { ( x - 1 ) ^ { 2 } } { 9 } + \frac { ( y - 4 ) ^ { 2 } } { 4 } = - 1$
d.
$$\frac { ( x - 1 ) ^ { 2 } } { 4 } + \frac { ( y - 4 ) ^ { 2 } } { 9 } = 1$$
e. none of these

Question 49

Find the equation of the circle with a radius of 5 and its center at the intersection of $5 x + y = 22$ and $5 x -$ $5 y = 10$ .
a. $( x - 4 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25$
b. $( x - 4 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 25$
c. $( x + 4 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 5$
d. $( x - 4 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 5$
e. $( x + 4 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25$

Question 50

Find the equation of the curve on which the point $P$ lies. The distance between point $P$ and the point ( 0 ,
13) is $\frac { 13 } { 7 }$ of the distance between point $P$ and the line $y = - 7$ .
a. $\quad 49 x ^ { 2 } - 120 y ^ { 2 } + 3,640 y = 0$
b. $\quad 49 x ^ { 2 } - 120 y ^ { 2 } - 3,640 y = 0$
c. $\quad 49 x ^ { 2 } - 120 y ^ { 2 } - 3,640 y = 69$
d. $\quad 3,640 x ^ { 2 } - 120 y ^ { 2 } - 49 y = 0$
e. none of these

Question 51

Find the equation of the circle graphed below.
a. $y ^ { 2 } = x ^ { 2 } + 4$
b. $\quad x ^ { 2 } + y ^ { 2 } = 2$
c. $x ^ { 2 } + y ^ { 2 } = 4$
d. $\quad x ^ { 2 } + y = 4$
e. $\quad x ^ { 2 } + y ^ { 2 } = 1$

Question 52

Stones dropped into a calm pond at points $A$ and $B$ create ripples that propagate in widening circles. In the figure below, points $A$ and $B$ are 12 feet apart, and the radii of the circles differ by 4 feet. The point $P ( x , y )$ where the circles intersect moves along a curve. Find the equation of the curve.


a. $\quad \frac { ( x - 5 ) ^ { 2 } } { 4 } - \frac { ( y - 4 ) ^ { 2 } } { 32 } = 1$
b. $\quad \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 32 } = 1$
c. $\quad \frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 64 } = 1$
d. $\quad \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 32 } = 1$
e. none of these

Question 53

Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l } x ^ { 2 } - 10 x - y = - 24 \\x ^ { 2 } - 10 x + y = - 24\end{array} \right.$$
a. $\quad ( 0,3 ) , ( 6,0 )$
b. $\quad ( 3,0 ) , ( 4,0 )$
c. $\quad ( 0,3 ) , ( 0,4 )$
d. $( 6,0 ) , ( 4,0 )$
e. no solution

Question 54

Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 20 \\x + y = 6\end{array} \right.$$
a. $\quad ( 2,4 ) , ( 4,2 )$
b. $\quad ( 4,4 ) , ( 4,4 )$
c. $\quad ( 2,4 ) , ( 2,4 )$
d. $( 4,2 ) , ( 4,2 )$
e. no solution

Question 55

Tell whether the parabolic graph of the equation opens up, down, to the left, or to the right. $$y ^ { 2 } = - 4 x$$
a. opens to the right
b. opens down
c. opens to the left
d. opens up

Question 56

The area of a rectangle is 24 square centimeters, and its perimeter is 20 centimeters.Find the dimensions of the rectangle. a. $\quad 6 \mathrm {~cm}$ by $7 \mathrm {~cm}$
b. $20 \mathrm {~cm}$ by $20 \mathrm {~cm}$
c. $\quad 6 \mathrm {~cm}$ by $6 \mathrm {~cm}$
d. $\quad 6 \mathrm {~cm}$ by $4 \mathrm {~cm}$
e. $\quad 7 \mathrm {~cm}$ by $4 \mathrm {~cm}$

Question 57

Find the equation of the parabola passing through the given points. $( - 2,3 ) , ( 1,12 )$ , and $( 2,19 )$
a. $\quad y = 4 x + 6$
b. $\quad y = x ^ { 2 } - 4 x + 7$
c. $\quad y = x ^ { 2 } + 4 x + 7$
d. $\quad y = - 5 x ^ { 2 } + 4 x + 31$
e. none of these

Question 58

Find the equation of the circle with a radius of 9 and its center at the intersection of $4 x + y = 1$ and $- 5 x -$ $4 y = 7$ .
a. $( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 9$
b. $( x - 1 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 81$
c. $( x + 1 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 9$
d. $( x + 1 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 81$
e. none of these

Question 59

Write the equation of the ellipse that has its center at the origin, focus at $( 1,0 )$ and $\frac { 7 } { 8 }$ is one-half the length of the minor axis.
a. $\quad \frac { 64 x ^ { 2 } } { 113 } + \frac { 64 y ^ { 2 } } { 49 } = 1$
b. $\quad \frac { 64 x ^ { 2 } } { 113 } - \frac { y ^ { 2 } } { 49 } = 1$
c. $\quad \frac { 8 x ^ { 2 } } { 7 } + \frac { 8 y ^ { 2 } } { 15 } = 1$
d. $\frac { x ^ { 2 } } { 113 } + \frac { y ^ { 2 } } { 49 } = 1$
e. $\frac { 64 x ^ { 2 } } { 113 } + \frac { 64 y ^ { 2 } } { 49 } = - 1$

Question 60

Find the equation of the parabola with vertex at (0, 0) and focus at (0, 3). a. $\quad y ^ { 2 } = 12 x$
b. $\quad x ^ { 2 } = y + 3$
c. $\quad x ^ { 2 } = 3 y$
d. $\quad x ^ { 2 } = 12 y$
e. none of these

Question 61

Stones dropped into a calm pond at points A and B create ripples that propagate in widening circles.In the figure below, points A and B are 12 feet apart, and the radii of the circles differ by 4 feet.The point
P (x, y) where the circles intersect moves along a curve.Find the equation of the curve.
a. $\quad \frac { ( x - 4 ) ^ { 2 } } { 4 } - \frac { ( y - 4 ) ^ { 2 } } { 32 } = 1$
b. $\quad \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 32 } = 1$
c. $\quad \frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 64 } = 1$
d. $\quad \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 32 } = 1$
e. none of these

Question 62

Solve the system of equations algebraically for real values of x and y.

A)(8, 3), (3, 8)
B)(7, 3), (3, 7)
C)(7, 3), (7, 8)
D)(3, 7), (3, 7)
E)no solution

Question 63

Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 65 \\y = x ^ { 2 } - 9\end{array} \right.$$
a. $\quad ( - 4,7 ) , ( 4,3 )$
b. $\quad ( 4,7 ) , ( 7 , - 4 )$
c. $\quad ( 7,4 ) , ( - 4,7 )$
d. $( 4,7 ) , ( - 4,7 )$
e. $\quad ( 3,7 ) , ( 7 , - 4 )$

Question 64

Identify the conic as a circle, parabola, ellipse, or hyperbola. $$6 x ^ { 2 } = 24 x + 2 \left( y ^ { 2 } - 8 y \right) + 20$$

A)circle
B)hyperbola
C)ellipse
D)parabola
E)none of the above

Question 65

Find the equation of the hyperbola with center $( 2,1 ) , a = 5$ and $b = 2$ . Transverse axis is horizontal.
a. $\quad \frac { ( x - 3 ) ^ { 2 } } { 23 } - \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
b. $\quad \frac { ( x - 2 ) ^ { 2 } } { 25 } - \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
c. $\quad \frac { ( x + 2 ) ^ { 2 } } { 25 } - \frac { ( y + 1 ) ^ { 2 } } { 4 } = 1$
d. $\quad \frac { ( x - 2 ) ^ { 2 } } { 25 } + \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1$
e. none of these

Question 66

An arch is a semi-ellipse 6 meters wide and 3 meters high.Write the equation of the ellipse if the ellipse is centered at the origin. a. $\quad \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 3 } = 1$
b. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 9 } = 1$
c. $\quad \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 9 } = 1$
d. $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 6 } = 1$
e. none of these

Question 67

Find the equation of the ellipse with center at $( 8,9 )$ , vertex at $( 16,9 )$ , and passing through the intersection of $y = - 2 x + 32$ and $y = x + 8$ .
a. $\quad \frac { ( x - 8 ) ^ { 2 } } { 8 } + \frac { ( y - 9 ) ^ { 2 } } { 7 } = 1$
b. $\quad \frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 9 ) ^ { 2 } } { 49 } = 1$
c. $\quad \frac { ( x - 8 ) ^ { 2 } } { 49 } + \frac { ( y - 9 ) ^ { 2 } } { 49 } = 1$
d. $\quad \frac { ( x - 8 ) ^ { 2 } } { 81 } + \frac { ( y - 9 ) ^ { 2 } } { 64 } = 1$
e.
$$\frac { ( x - 8 ) ^ { 2 } } { 49 } + \frac { ( y - 9 ) ^ { 2 } } { 64 } = 1$$

Question 68

Find the area of the fundamental rectangle of the hyperbola. $$64 ( x - 3 ) ^ { 2 } - 36 ( y - 5 ) ^ { 2 } = 2,304$$
a. 192 square units
b. $\quad 96$ square units
c. $\quad 384$ square units
d. $\quad 48$ square units
e. none of these

Question 69

Parallel beams of similarly charged particles are shot from two atomic accelerators $L = 34$ meters apart, as shown in the figure below. If the particles were not deflected, the beams would be $H = 3.9 \times 10 ^ { - 4 }$ meters apart. However, because the charged particles repel each other, the beams follow the hyperbolic path $y = \frac { k } { x }$ , for some $k$ . Find $k$ .


a. $\quad k = 3.315 \times 10 ^ { - 3 }$
b. $\quad k = 6.63 \times 10 ^ { - 3 }$
c. $\quad k = 3.315$
d. $\quad k = 3.315 \times 10 ^ { - 6 }$
e. none of these

Question 70

The area of a rectangle is 36 square centimeters, and its perimeter is 26 centimeters.Find the dimensions of the rectangle.

A)4 cm by 7 cm
B)7 cm by 9 cm
C)26 cm by 26 cm
D)4 cm by 4 cm
E)4 cm by 9 cm

Question 71

Solve the system of equations algebraically for real values of x and y. $$\left\{ \begin{array} { l } x ^ { 2 } - 8 x - y = - 7 \\x ^ { 2 } - 8 x + y = - 7\end{array} \right.$$
a. $\quad ( 0,4 ) , ( 0,1 )$
b. $\quad ( 0,4 ) , ( 7,0 )$
c. $\quad ( 4,0 ) , ( 1,0 )$
d. $( 7,0 ) , ( 1,0 )$
e. no solution

Question 72

Find the graph of the following equation. $$\frac { ( x - 3 ) ^ { 2 } } { 9 } + \frac { 9 y ^ { 2 } } { 25 } = 1$$
a.


b.

c.

Question 73

Jim drove 252 miles. Jim's brother made the same trip at a speed 14 miles per hour slower than Jim did and required an extra $1 \frac { 1 } { 2 }$ hours. Find Jim's rate and time.
a. $\quad 56 \mathrm { mph } , 3.5 \mathrm { hr }$
b. $\quad 72 \mathrm { mph } , 4.5 \mathrm { hr }$
c. $\quad 72 \mathrm { mph } , 3.5 \mathrm { hr }$
d. $\quad 56 \mathrm { mph } , 4.5 \mathrm { hr }$
e. not possible

Question 74

Find the equation of the hyperbola with center $( 1 , - 3 ) , a ^ { 2 } = 36$ and $b ^ { 2 } = 16$
a. $\quad \frac { ( x - 1 ) ^ { 2 } } { 36 } - \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$
b. $\quad \frac { ( x - 1 ) ^ { 2 } } { 36 } - \frac { ( y - 3 ) ^ { 2 } } { 16 } = 1$
c. $\quad \frac { ( x + 1 ) ^ { 2 } } { 36 } - \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$
d. $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 16 } = 1$
e. $\quad \frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y + 3 ) ^ { 2 } } { 36 } = 1$

Question 75

Find the equation of the curve on which the point $P$ lies. The distance between point $P$ and the point $( 0$ , 7) is $\frac { 7 } { 4 }$ of the distance between point $P$ and the line $y = - 4$ .
a. $\quad 616 x ^ { 2 } - 33 y ^ { 2 } - 16 y = 0$
b. $\quad 16 x ^ { 2 } - 33 y ^ { 2 } - 616 y = 59$
c. $\quad 16 x ^ { 2 } - 33 y ^ { 2 } - 616 y = 0$
d. $\quad 16 x ^ { 2 } - 33 y ^ { 2 } + 616 y = 0$
e. none of these