Exam 12: The Conic Sections

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

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

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

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

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. $r = \frac { 13 } { 5 - 5 \sin \theta }$

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

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

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

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

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

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 $\overline { P Q }$ .

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.

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. $\theta = 120 ^ { \circ }$ $\left( x ^ { \prime } , y ^ { \prime } \right) = ( \sqrt { 3 } , 8 )$

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

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

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. $175 x y - 600 y ^ { 2 } - 1 = 0$

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

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

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

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