Exam 12: The Conic Sections

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. $\frac { x ^ { 2 } } { 87 ^ { 2 } } + \frac { y ^ { 2 } } { 29 ^ { 2 } } = 1 ; P ( 63,20 )$

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

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

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. $124 x ^ { 2 } + 7 x y + 100 y ^ { 2 } = 0$

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