Exam 11: Conic Sections

Find the focus, directrix, and focal diameter of the parabola $8 x + 5 y ^ { 2 } = 0$ , and sketch its graph.

Write a polar equation of a hyperbola with eccentricity $6$ and directrix $x = - 3$ .

Find an equation for the ellipse with major axis of length $12$ , minor axis of length $8$ , foci on the $x$ -axis, and centered at the origin.

Find the vertices, foci, and eccentricity of the ellipse given by $x ^ { 2 } + 9 y ^ { 2 } = 1$ . Determine the lengths of the major and minor axes, and sketch the graph.

Find the equation for the hyperbola that has asymptotes $y = \pm 3 x$ and passes through the point $( 3 , - 6 )$ .

Find the eccentricity and directrix of the conic $r = \frac { 6 } { 2 + 2 \sin \theta }$ and graph the conic. Find an equation for the conic rotated clockwise about its vertex through an angle $3 \pi / 2$ , and graph it.

Find an equation for a parabola with vertex at the origin, and with focus $F ( 0,5 / 2 )$ .

A conic has equation $6 x ^ { 2 } + 3 y ^ { 2 } + x - 3 y = 1 - 3 \sqrt { 3 } x y$ . (a) Use the discriminant to identify the conic. (a) by graphing the conic with a graphing calculator. (b) Confirm your answer in part (c) Find the angle necessary to eliminate the $x y$ - term.

Find the eccentricity and directrix of the conic $r = \frac { 3 } { 1 - \cos \theta / 2 }$ and graph the conic. Find an equation for the conic rotated counterclockwise about the origin through an angle $\pi / 6$ , and graph it.

Complete the square to determine whether the equation $y ^ { 2 } - 2 y - 4 x ^ { 2 } - 16 x = 15$ represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph, explain why.

Find an equation for the ellipse whose graph is shown.

Find an equation for the hyperbola that has foci $( 0 , \pm 10 )$ and vertices $( 0 , \pm 3 )$ .

Find the vertex, focus, and directrix of the parabola $( y - 3 ) ^ { 2 } = 12 ( x + 1 )$ , and sketch its graph.

Find the eccentricity and identify the conic given by $r = \frac { 6 } { 5 - 2 \sin \theta }$ , sketch it and label its vertices.

Determine the equation of the conic $y = ( x + 2 ) ^ { 2 }$ in $X Y$ - coordinates if the axes are rotated through an angle $\phi = 30 ^ { \circ }$ .

Find the focus, directrix, and focal diameter of the parabola $3 x ^ { 2 } + 5 y = 0$ , and sketch its graph.

Find an equation for the hyperbola that has foci $( \pm 3,0 )$ and passes through the point $( 4,7 )$ .

Find an equation for the ellipse with foci $( 0 , \pm 3 )$ , and major axis of length $10$ .

Find the vertices, foci, and eccentricity of the ellipse given by $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ . Determine the lengths of the major and minor axes, and sketch the graph.

Find the focus, directrix, and focal diameter of the parabola $y ^ { 2 } = 8 x$ , and sketch its graph.