Deck 12: The Conic Sections

Question 1

Use the given information to find the equation of the hyperbola. The foci are $( \pm 6,0 )$ and the directrices are $x = \pm 3$

A) $9 x ^ { 2 } - 5 y ^ { 2 } = 18$
B) $9 x ^ { 2 } - y ^ { 2 } = 61$
C) $x ^ { 2 } - 5 y ^ { 2 } = 61$
D) $x ^ { 2 } - y ^ { 2 } = 18$
E) $2 x ^ { 2 } - y ^ { 2 } = 18$

Answer 1

$x ^ { 2 } - y ^ { 2 } = 18$

Question 2

Determine the directrices for the ellipse and hyperbola. $64 x ^ { 2 } + 81 y ^ { 2 } = 5,184,64 x ^ { 2 } - 81 y ^ { 2 } = 5,184$

A) $\text { ellipse directrices: } x = \pm 9 \sqrt { 17 } \text {; hyperbola directrices: } x = \pm 9 \sqrt { 145 }$
B) $\text { ellipse directrices: } x = \pm \frac { 81 \sqrt { 17 } } { 17 } ; \text { hyperbola directrices: } x = \pm \frac { 81 \sqrt { 145 } } { 145 }$
C) $\text { ellipse directrices: } x = \pm \frac { 81 \sqrt { 145 } } { 17 } ; \text { hyperbola directrices: } x = \pm \frac { 81 \sqrt { 17 } } { 145 }$
D) $\text { ellipse directrices: } x = - 9 \sqrt { 17 } \text {; hyperbola directrices: } x = 9 \sqrt { 145 }$
E) $\text { ellipse directrices: } x = \frac { 9 \sqrt { 17 } } { 17 } ; \text { hyperbola directrices: } x = - \frac { 81 \sqrt { 145 } } { 145 }$

Answer 2

$\text { ellipse directrices: } x = \pm \frac { 81 \sqrt { 17 } } { 17 } ; \text { hyperbola directrices: } x = \pm \frac { 81 \sqrt { 145 } } { 145 }$

Question 3

Graph the hyperbola. Specify the following: center, vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. $x ^ { 2 } - y ^ { 2 } - 2 x + 4 y - 19 = 0$

A) center: $( - 1 , - 2 )$ ; vertices: $( - 5 , - 2 ) , ( 3 , - 2 )$ ;
Foci: $( - 1 \pm 4 \sqrt { 2 } , - 2 )$ ;
Length of transverse axis: 8;
Length of conjugate axis: 8;
Eccentricity: $\sqrt { 2 }$ ;
Asymptotes: $y = - x - 3 , y = x - 1$ .
B)
Center: $( - 1 , - 2 )$ ;
Vertices: $( - 1 , - 6 ) , ( - 1,2 )$ ;
Foci: $( - 1 , - 2 \pm 4 \sqrt { 2 } )$ ;
length of transverse axis: 8;
Length of conjugate axis: 8;
Eccentricity: $\sqrt { 2 }$ ;
Asymptotes: $y = - x - 3 , y = x - 1$ .
C) center: $( 1,2 )$ ; vertices: $( 1 , - 2 ) , ( 1,6 )$ ;
Foci: $( 1,2 \pm 4 \sqrt { 2 } )$ ;
Length of transverse axis: 4;
Length of conjugate axis: 4;
Eccentricity: $2$ ;
Asymptotes: $y = - x + 3 , y = x + 1$ .
D) center: $( 1,2 )$ ; vertices: $( 1 , - 2 ) , ( 1,6 )$ ;
Foci: $( 1,2 \pm 4 \sqrt { 2 } )$ ;
Length of transverse axis: 8;
Length of conjugate axis: 8;
Eccentricity: $\sqrt { 2 }$ ;
Asymptotes: $y = - x + 3 , y = x + 1$ .
E) center: $( 1,2 )$ ; vertices: $( - 3,2 ) , ( 5,2 )$ ;
Foci: $( 1 \pm 4 \sqrt { 2 } , 2 )$ ;
Length of transverse axis: 8;
Length of conjugate axis: 8;
Eccentricity: $\sqrt { 2 }$ ;
Asymptotes: $y = - x + 3 , y = x + 1$ .

Answer 3

center: $( 1,2 )$ ; vertices: $( - 3,2 ) , ( 5,2 )$ ;
Foci: $( 1 \pm 4 \sqrt { 2 } , 2 )$ ;
Length of transverse axis: 8;
Length of conjugate axis: 8;
Eccentricity: $\sqrt { 2 }$ ;
Asymptotes: $y = - x + 3 , y = x + 1$ .

Question 4

Graph the ellipse. Specify the lengths of the major and minor axes, the foci, the center and the eccentricity. $\frac { ( x - 2 ) ^ { 2 } } { 2 ^ { 2 } } + \frac { ( y + 5 ) ^ { 2 } } { 3 ^ { 2 } } = 1$

A) center: (- 2, 5);
Length of major axis: 6;
Length of minor axis: 4;
Foci: $( - 2,5 \pm \sqrt { 5 } )$ ;
Eccentricity: $\frac { \sqrt { 5 } } { 3 }$
B) center: (1, - 2.5);
Length of major axis: 3;
Length of minor axis: 2;
Foci: $\left( 1 \pm \frac { \sqrt { 5 } } { 2 } , - 2.5 \right)$ ;
Eccentricity: $\frac { \sqrt { 5 } } { 3 }$
C) center: (2, - 5);
Length of major axis: 6;
Length of minor axis: 4;
Foci: $( 2 , - 5 \pm \sqrt { 5 } )$ ;
Eccentricity: $\frac { \sqrt { 5 } } { 3 }$ $$\theta$$
D) center: (1, - 2.5);
Length of major axis: 3;
Length of minor axis: 2;
Foci: $\left( 2 , - 1.25 \pm \frac { \sqrt { 5 } } { 2 } \right)$ ;
Eccentricity: $\frac { \sqrt { 5 } } { 3 }$
E) center: (2, - 5);
Length of major axis: 6;
Length of minor axis: 4;
Foci: $( 2 \pm \sqrt { 5 } , - 5 )$ ;
Eccentricity: $\frac { \sqrt { 5 } } { 3 }$

Question 5

Graph the parabola. Specify the focus, directrix, vertex and focal width. $x ^ { 2 } + 5 y + 25 = 0$

A) Focal width: 1
B) Focal width: 5
C) Focal width: 5
D) Focal width: 5
E) Focal width: 1

Question 6

Find the equation of the tangent to the parabola at the given point. $x ^ { 2 } = 8 y , ( 8,8 )$

A) y = 2x - 16
B) y = 3x - 9
C) y = 2x - 9
D) y = 3x - 8
E) y = 2x - 8

Question 7

Graph the hyperbola. Specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. $x ^ { 2 } - 25 y ^ { 2 } = 25$

A) vertices: $( \pm 5,0 )$ ;
Foci: $( \pm \sqrt { 26 } , 0 )$ ;
Length of transverse axis: 10;
Length of conjugate axis: 2;
Eccentricity: $\frac { \sqrt { 26 } } { 5 }$
Asymptotes: $y = \pm \frac { 1 } { 5 } x$ .
B) vertices: $( 0 , \pm 6 )$ ;
Foci: $( 0 , \pm \sqrt { 37 } )$ ;
Length of transverse axis: 2;
Length of conjugate axis: 12;
Eccentricity: $\sqrt { 37 }$ ;
Asymptotes: $y = \pm \frac { 1 } { 6 } x$ .
C) vertices: $( \pm 1,0 )$ ;
Foci: $( \pm \sqrt { 26 } , 0 )$ ;
Length of transverse axis: 2;
Length of conjugate axis: 10;
Eccentricity: $\sqrt { 26 }$ ;
Asymptotes: $y = \pm 5 x$ .
D) vertices: $( 0 , \pm 5 )$ ;
Foci: $( 0 , \pm \sqrt { 26 } )$ ;
Length of transverse axis: 10;
Length of conjugate axis: 2;
Eccentricity: $\frac { \sqrt { 26 } } { 5 }$
Asymptotes: $y = \pm 5 x$ .
E) vertices: $( 0 , \pm 1 )$ ;
Foci: $( 0 , \pm \sqrt { 26 } )$ ;
Length of transverse axis: 2;
Length of conjugate axis: 10;
Eccentricity: $\sqrt { 26 }$ ;
Asymptotes: $y = \pm \frac { 1 } { 5 } x$ .

Question 8

Graph the hyperbola. Specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. $3 y ^ { 2 } - 7 x ^ { 2 } = 1$

A) vertices: $\left( \pm \frac { \sqrt { 7 } } { 7 } , 0 \right)$ ;
Foci: $\left( \pm \frac { \sqrt { 210 } } { 21 } , 0 \right)$ ;
Length of transverse axis: $\frac { 2 \sqrt { 3 } } { 3 }$ ;
Length of conjugate axis: $\frac { 2 \sqrt { 7 } } { 7 }$ ;
Eccentricity: $\frac { \sqrt { 70 } } { 7 }$ ;
Asymptotes: $y = \pm \frac { \sqrt { 21 } } { 3 } x$ .
B) vertices: $\left( \pm \frac { \sqrt { 3 } } { 6 } , 0 \right)$ ;
Foci: $\left( \pm \frac { 2 \sqrt { 3 } } { 3 } , 0 \right)$ ;
Length of transverse axis: $\frac { \sqrt { 3 } } { 3 }$ ;
Length of conjugate axis: $2$ ;
Eccentricity: $2$ ;
Asymptotes: $y = \pm \frac { \sqrt { 3 } } { 3 } x$ .
C)
vertices: $\left( 0 , \pm \frac { \sqrt { 3 } } { 3 } \right)$ ;
Foci: $\left( 0 , \pm \frac { \sqrt { 210 } } { 21 } \right)$ ;
Length of transverse axis: $\frac { 2 \sqrt { 3 } } { 3 }$ ;
Length of conjugate axis: $\frac { 2 \sqrt { 7 } } { 7 }$ ;
Eccentricity: $\frac { \sqrt { 70 } } { 7 }$ ;
Asymptotes: $y = \pm \frac { \sqrt { 21 } } { 7 } x$ .
D) vertices: $\left( \pm \frac { \sqrt { 7 } } { 7 } , 0 \right)$ ;
Foci: $\left( \pm \frac { \sqrt { 210 } } { 21 } , 0 \right)$ ;
Length of transverse axis: $\frac { 2 \sqrt { 7 } } { 7 }$ ;
Length of conjugate axis: $\frac { 2 \sqrt { 3 } } { 3 }$ ;
Eccentricity: $\frac { \sqrt { 70 } } { 7 }$ ;
Asymptotes: $y = \pm \frac { \sqrt { 21 } } { 7 } x$ .
E) vertices: $\left( 0 , \pm \frac { \sqrt { 3 } } { 3 } \right)$ ;
Foci: $\left( 0 , \pm \frac { \sqrt { 210 } } { 21 } \right)$ ;
length of transverse axis: $\frac { 2 \sqrt { 3 } } { 3 }$ ;
Length of conjugate axis: $\frac { 2 \sqrt { 7 } } { 7 }$ ;
Eccentricity: $\frac { \sqrt { 70 } } { 7 }$ ;
Asymptotes: $y = \pm \frac { \sqrt { 21 } } { 3 } x$ .

Question 9

Select the answer that represents the hyperbola as well as its eccentricity, center and values of $a$ , $b$ and $c$ . $r = \frac { 5 } { 2 + 6 \cos \theta }$

A) Eccentricity: $3$ Center: $\left( \frac { 15 } { 16 } , 0 \right)$ $a = \frac { 5 } { 16 } , b = \frac { 5 \sqrt { 2 } } { 8 } , c = \frac { 15 } { 16 }$
B) Eccentricity: $3$ Center: $\left( - \frac { 15 } { 16 } , 0 \right)$ $a = \frac { 5 } { 16 } , b = \frac { 5 \sqrt { 2 } } { 8 } , c = \frac { 15 } { 16 }$
C) Eccentricity: $3$ Center: $\left( \frac { 1 } { 2 } , 0 \right)$ $a = \frac { 1 } { 6 } , b = \frac { \sqrt { 2 } } { 3 } , c = \frac { 1 } { 2 }$
D) Eccentricity: $3$ Center: $\left( - \frac { 1 } { 2 } , 0 \right)$ $a = \frac { 1 } { 6 } , b = \frac { \sqrt { 2 } } { 3 } , c = \frac { 1 } { 2 }$
E) Eccentricity: $3$ Center: $\left( \frac { 15 } { 16 } , 0 \right)$ $a = \frac { 1 } { 6 } , b = \frac { \sqrt { 2 } } { 3 } , c = \frac { 1 } { 2 }$

Question 10

You are given an ellipse and a point P on the ellipse. Find $F _ { 1 } P$ and $F _ { 2 } P$ , the lengths of the focal radii. $\frac { x ^ { 2 } } { 87 ^ { 2 } } + \frac { y ^ { 2 } } { 29 ^ { 2 } } = 1 ; P ( 63,20 )$

A) $F _ { 1 } P = 87 + 42 \sqrt { 2 } , F _ { 2 } P = 87 - 42 \sqrt { 2 }$
B) $F _ { 1 } P = 841 + 42 \sqrt { 2 } , F _ { 2 } P = 29 + 42 \sqrt { 2 }$
C) $F _ { 1 } P = 29 + 43 \sqrt { 2 } , F _ { 2 } P = 841 - 43 \sqrt { 2 }$
D) $F _ { 1 } P = - 87 - 42 \sqrt { 2 } , F _ { 2 } P = 87 + 42 \sqrt { 3 }$
E) $F _ { 1 } P = - 29 + 43 \sqrt { 2 } , F _ { 2 } P = - 841 - 43 \sqrt { 2 }$

Question 11

Graph the hyperbola. Specify the following: center, vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. $9 x ^ { 2 } - 18 x - 16 y ^ { 2 } + 96 y - 279 = 0$

A)
B)
C)
D)
E)

Question 12

Suppose the line $x - 7 y + 55 = 0$ intersects the circle $x ^ { 2 } - 10 x + y ^ { 2 } - 10 y = - 25$ at points $P$ and $e$ . Find the length of the chord $\overline { P Q }$ .

A) $\sqrt { 5 }$
B) $5 \sqrt { 2 }$
C) $2 \sqrt { 5 }$
D) $36 \sqrt { 2 }$
E) $6 \sqrt { 2 }$

Question 13

The point $( 1 , - 2 )$ is the midpoint of a chord of the circle $x ^ { 2 } - 4 x + y ^ { 2 } + 2 y = 8$ . Find the length of the chord.

A) $4 \sqrt { 11 }$
B) $\sqrt { 22 }$
C) $22$
D) $11 \sqrt { 2 }$
E) $2 \sqrt { 11 }$

Question 14

Graph the ellipse. Specify the lengths of the major and minor axes, the foci and the eccentricity. $9 x ^ { 2 } + 16 y ^ { 2 } = 144$

A) length of major axis: 8;
Length of minor axis: 6;
Foci: $( 0 , \pm \sqrt { 7 } )$ ;
Eccentricity: $\frac { \sqrt { 7 } } { 4 }$
B) length of major axis: 4;
Length of minor axis: 3;
Foci: $( 0 , \pm \sqrt { 7 } )$ ;
Eccentricity: $\frac { \sqrt { 7 } } { 4 }$
C) length of major axis: 9.6;
Length of minor axis: 7.2;
Foci: $( 0 , \pm \sqrt { 7 } )$ ;
Eccentricity: $$\frac { \sqrt { 7 } } { 4 }$$
D) length of major axis: 8;
Length of minor axis: 6;
Foci: $( \pm \sqrt { 7 } , 0 )$ ;
Eccentricity: $\frac { \sqrt { 7 } } { 4 }$
E) length of major axis: 4;
Length of minor axis: 3;
Foci: $( \pm \sqrt { 7 } , 0 )$ ;
Eccentricity: $\frac { \sqrt { 7 } } { 4 }$

Question 15

Find the equation of the tangent to the parabola at the given point. $x ^ { 2 } = - y , ( - 3 , - 9 )$

A) y = 5x + 8
B) y = 6x + 8
C) y = 6x + 9
D) y = 6x + 18
E) y = 5x + 9

Question 16

Graph the hyperbola. Specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. $3 y ^ { 2 } - 11 x ^ { 2 } = 1$

A) vertices: $\left( \pm \frac { \sqrt { 11 } } { 11 } , 0 \right)$ ;
Foci: $\left( \pm \frac { \sqrt { 462 } } { 33 } , 0 \right)$ ;
Length of transverse axis: $\frac { 2 \sqrt { 11 } } { 11 }$ ;
Length of conjugate axis: $\frac { 2 \sqrt { 3 } } { 3 }$ ;
Eccentricity: $\frac { \sqrt { 154 } } { 11 }$ ;
Asymptotes: $y = \pm \frac { \sqrt { 33 } } { 11 } x$ .
B) vertices: $\left( \pm \frac { \sqrt { 11 } } { 11 } , 0 \right)$ ;
Foci: $\left( \pm \frac { \sqrt { 462 } } { 33 } , 0 \right)$ ;
Length of transverse axis: $\frac { 2 \sqrt { 3 } } { 3 }$ ;
Length of conjugate axis: $\frac { 2 \sqrt { 11 } } { 11 }$ ;
Eccentricity: $\frac { \sqrt { 154 } } { 11 }$ ;
Asymptotes: $y = \pm \frac { \sqrt { 33 } } { 3 } x$ .
C) vertices: $\left( 0 , \pm \frac { \sqrt { 3 } } { 3 } \right)$ ;
Foci: $\left( 0 , \pm \frac { \sqrt { 462 } } { 33 } \right)$ ;
Length of transverse axis: $\frac { 2 \sqrt { 3 } } { 3 }$ ;
Length of conjugate axis: $\frac { 2 \sqrt { 11 } } { 11 }$ ;
Eccentricity: $\frac { \sqrt { 154 } } { 11 }$ ;
Asymptotes: $y = \pm \frac { \sqrt { 33 } } { 3 } x$ .
D) vertices: $\left( \pm \frac { \sqrt { 3 } } { 6 } , 0 \right)$ ;
Foci: $\left( \pm \frac { 2 \sqrt { 3 } } { 3 } , 0 \right)$ ;
Length of transverse axis: $\frac { \sqrt { 3 } } { 3 }$ ;
Length of conjugate axis: $2$ ;
Eccentricity: $2$ ;
Asymptotes: $y = \pm \frac { \sqrt { 3 } } { 3 } x$ .
E) vertices: $\left( 0 , \pm \frac { \sqrt { 3 } } { 3 } \right)$ ;
Foci: $\left( 0 , \pm \frac { \sqrt { 462 } } { 33 } \right)$ ;
Length of transverse axis: $\frac { 2 \sqrt { 3 } } { 3 }$ ;
Length of conjugate axis: $\frac { 2 \sqrt { 11 } } { 11 }$ ;
Eccentricity: $\frac { \sqrt { 154 } } { 11 }$ ;
Asymptotes: $y = \pm \frac { \sqrt { 33 } } { 11 } x$ .

Question 17

Find the center and the radius of the circle that passes through the points $( - 4,12 ) , ( 10 , - 2 ) \text { and } ( 2 , - 6 )$

A) $\text { The center is at } ( - 3,2 ) \text { and the radius is } 2 .$
B) The center is at $( 0,2 )$ and the radius is $0 .$
C) The center is at $( 3,3 )$ and the radius is $1 .$
D) The center is at $( 0,0 )$ and the radius is 7.
E) The center is at $( 2,4 )$ and the radius is $10 .$

Question 18

Graph the hyperbola. Specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. $y ^ { 2 } - 4 x ^ { 2 } = 4$

A) vertices: $( 0 , \pm 2 )$ ;
Foci: $( 0 , \pm \sqrt { 5 } )$ ;
Length of transverse axis: 4;
Length of conjugate axis: 2;
Eccentricity: $\frac { \sqrt { 5 } } { 2 }$
Asymptotes: $y = \pm 2 x$ .
B) vertices: $( 0 , \pm 1 )$ ;
Foci: $( 0 , \pm \sqrt { 5 } )$ ;
Length of transverse axis: 2;
Length of conjugate axis: 4;
Eccentricity: $\sqrt { 5 }$ ;
Asymptotes: $y = \pm \frac { 1 } { 2 } x$ .
C) vertices: $( \pm 2,0 )$ ;
Foci: $( \pm \sqrt { 5 } , 0 )$ ;
Length of transverse axis: 4;
Length of conjugate axis: 2;
Eccentricity: $\frac { \sqrt { 5 } } { 2 }$
Asymptotes: $y = \pm \frac { 1 } { 2 } x$ .
D) vertices: $( \pm 1,0 )$ ;
Foci: $( \pm \sqrt { 37 } , 0 )$ ;
Length of transverse axis: 2;
Length of conjugate axis: 12;
Eccentricity: $\sqrt { 37 }$ ;
Asymptotes: $y = \pm 37 x$ .
E) vertices: $( \pm 1,0 )$ ;
Foci: $( \pm \sqrt { 5 } , 0 )$ ;
Length of transverse axis: 2;
Length of conjugate axis: 4;
Eccentricity: $$\sqrt { 5 }$$ Asymptotes: $y = \pm 2 x$ .

Question 19

Use the given information to find the equation of the ellipse. The foci are $( \pm 2,0 )$ and the directrices are $x = \pm 3$ .

A) $5 x ^ { 2 } + 3 y ^ { 2 } = 14$
B) $5 x ^ { 2 } + 5 y ^ { 2 } = 6$
C) $x ^ { 2 } + 5 y ^ { 2 } = 6$
D) $x ^ { 2 } + 3 y ^ { 2 } = 6$
E) $x ^ { 2 } + 5 y ^ { 2 } = 14$

Question 20

Graph the ellipse. Specify the lengths of the major and minor axes, the foci, the center and the eccentricity. $4 x ^ { 2 } + 25 y ^ { 2 } - 250 y + 525 = 0$

A) center: (0, 5);
Degenerate ellipse
B) center: (1, $\frac { 3 } { 2 }$ );
Degenerate ellipse
C) center: (5,0);
Length of major axis: 10;
Length of minor axis: 4;
Foci: $( 5 , \pm \sqrt { 21 } )$ ;
Eccentricity: $\frac { \sqrt { 21 } } { 5 }$
D) center: (0, 5);
Length of major axis: 10;
Length of minor axis: 4;
Foci: $( 0,5 \pm \sqrt { 21 } )$ ;
Eccentricity: $\frac { \sqrt { 21 } } { 5 }$
E) center: (0, 5);
Length of major axis: 10;
Length of minor axis: 4;
Foci: $( \pm \sqrt { 21 } , 5 )$ ;
Eccentricity: $\frac { \sqrt { 21 } } { 5 }$

Question 21

An angle of rotation is specified, followed by the coordinates of a point in the $x ^ { \prime } - y ^ { \prime }$ system. Find the coordinates of the point with respect to the $x - y$ system. $\theta = 120 ^ { \circ }$ $\left( x ^ { \prime } , y ^ { \prime } \right) = ( \sqrt { 3 } , 8 )$

A) $\left( \frac { 5 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$
B) $\left( - \frac { 9 \sqrt { 3 } } { 2 } , - \frac { 5 } { 2 } \right)$
C) $\left( - \frac { 5 } { 2 } , - \frac { 9 \sqrt { 3 } } { 2 } \right)$
D) $\left( \frac { 9 \sqrt { 3 } } { 2 } , - \frac { 5 } { 2 } \right)$
E) $\left( \frac { 9 \sqrt { 3 } } { 2 } , \frac { 5 } { 2 } \right)$

Question 22

Select the graph that represents the given conic section. If the conic is a parabola, specify (using rectangular coordinates) the vertex and directrix. If the conic is an ellipse, specify the center, the eccentricity, and the lengths of the major and minor axes. If the conic is a hyperbola, specify the center, the eccentricity, and the lengths of the transverse and conjugate axes. $r = \frac { 13 } { 5 - 5 \sin \theta }$

A) Vertex: $\left( 0 , \frac { 13 } { 10 } \right)$ Directrix: $y = \frac { 13 } { 5 }$
B) Vertex: $\left( - \frac { 13 } { 10 } , 0 \right)$ Directrix: $x = - \frac { 13 } { 5 }$
C) Center: $( 0,0 )$ Eccentricity: $\frac { 5 } { 13 }$ Major axis length: $\frac { 26 } { 5 }$
Minor axis length: $\frac { 10 } { 13 }$
D) Vertex: $\left( \frac { 13 } { 10 } , 0 \right)$ Directrix: $x = \frac { 13 } { 5 }$
E) Vertex: $\left( 0 , - \frac { 13 } { 10 } \right)$ Directrix: $y = - \frac { 13 } { 5 }$

Question 23

Find $\sin \theta$ and $\cos \theta$ , where $\theta$ is the (acute) angle of rotation that eliminates the $x ^ { \prime } y ^ { \prime }$ -term. Note: You are not asked to graph the equation. $175 x y - 600 y ^ { 2 } - 1 = 0$

A) $\sin \theta = \frac { 4 } { 5 }$ $\cos \theta = \frac { 3 } { 5 }$
B) $\sin \theta = \frac { 7 \sqrt { 2 } } { 5 }$ $\cos \theta = \frac { \sqrt { 2 } } { 5 }$
C) $\sin \theta = \frac { 3 } { 5 }$ $\cos \theta = \frac { 4 } { 5 }$
D) $\sin \theta = \frac { \sqrt { 2 } } { 10 }$ $\cos \theta = \frac { 7 \sqrt { 2 } } { 10 }$
E) $\sin \theta = \frac { \sqrt { 2 } } { 2 }$ $\cos \theta = \frac { \sqrt { 2 } } { 2 }$

Question 24

Find $\sin \theta$ and $\cos \theta$ , where $\theta$ is the (acute) angle of rotation that eliminates the $x ^ { \prime } y ^ { \prime }$ -term. Note: You are not asked to graph the equation. $124 x ^ { 2 } + 7 x y + 100 y ^ { 2 } = 0$

A) $\sin \theta = \frac { 7 \sqrt { 2 } } { 26 }$ $\cos \theta = \frac { 17 \sqrt { 2 } } { 26 }$
B) $\sin \theta = \frac { \sqrt { 2 } } { 10 }$ $\cos \theta = \frac { 17 \sqrt { 2 } } { 26 }$
C) $\sin \theta = \frac { \sqrt { 2 } } { 10 }$ $\cos \theta = \frac { 7 \sqrt { 2 } } { 10 }$
D) $\sin \theta = \frac { 17 \sqrt { 2 } } { 26 }$ $\cos \theta = \frac { 7 \sqrt { 2 } } { 26 }$
E) $\sin \theta = \frac { 7 \sqrt { 2 } } { 10 }$ $\cos \theta = \frac { \sqrt { 2 } } { 10 }$

Question 25

Determine the graph that represents the equation. $5 x ^ { 2 } + 6 x y + 5 y ^ { 2 } - 10 \sqrt { 2 } x - 6 \sqrt { 2 } y + 8 = 0$

A)
B)
C)
D)
E) no graph