Exam 11: Conic Sections

Find an equation for the ellipse with foci $( \pm 7,0 )$ , and major axis of length $18$ .

Find an equation for the hyperbola whose graph is shown.

Find the vertices, foci, and eccentricity of the ellipse given by $\frac { 1 } { 3 } x ^ { 2 } + \frac { 1 } { 27 } y ^ { 2 } = \frac { 1 } { 9 }$ . Determine the lengths of the major and minor axes, and sketch the graph.

Find an equation for the hyperbola that has vertices $( \pm 3,0 )$ and asymptotes $y = \pm 5 x$ .

Find the vertices, foci, and eccentricity of the ellipse given by $25 x ^ { 2 } + y ^ { 2 } = 25$ . Determine the lengths of the major and minor axes, and sketch the graph.

Find the center, foci, vertices, and asymptotes of the hyperbola $\frac { ( x - 2 ) ^ { 2 } } { 9 } - \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$ , and sketch its graph.

Find the eccentricity and identify the conic given by $r = \frac { 4 } { 4 + \frac { 4 } { 3 } \cos \theta }$ , sketch it and label its vertices.

Find the center, foci, vertices, and asymptotes of the hyperbola $16 ( y + 1 ) ^ { 2 } - 4 ( x + 1 ) ^ { 2 } = 64$ , and sketch its graph.

Find an equation for the conic whose graph is shown.

Use the discriminant to determine if the graph of the equation, $3 x ^ { 2 } + 10 x y + 3 y ^ { 2 } + 8 = 0$ is a parabola, an ellipse or a hyperbola, then use a rotation of axes to eliminate the $x y$ - term, and sketch the graph.

Complete the square to determine whether the equation $\frac { 1 } { 36 } x ^ { 2 } - \frac { 1 } { 2 } x + \frac { 153 } { 50 } + \frac { 1 } { 25 } y ^ { 2 } - \frac { 9 } { 25 } y = 1$ represents an ellipse, parabola, hyperbola or 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 axis. 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.

A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 ft from the cannon and the highest point it reaches is 2400 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.

Find the vertices, foci, and asymptotes of the hyperbola $y ^ { 2 } - x ^ { 2 } = 1$ , and sketch its graph.

Find the eccentricity and identify the conic given by $r = \frac { 16 } { 2 + 6 \sin \theta }$ , sketch it and label its vertices.

A cable is suspended between two utility poles (the shape of the suspended cable is parabolic). If the poles are $324$ ft apart, and the lowest point of the cable is $18$ ft. below the top of the poles, find the parabolic equation of the suspended cable, placing the origin of the coordinate system at the vertex.

Find an equation of the parabola whose graph is shown.

A conic has equation $8 x y = 1 - x ^ { 2 } - y ^ { 2 }$ . (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 an equation of a parabola with horizontal axis and vertex at the origin that passes through the point $( - 1,5 )$ .

A conic has equation $17 x ^ { 2 } - 24 x y + 10 y ^ { 2 } + x - 15 = - 4 y$ . (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 an equation of a parabola with vertex at the origin, focus on the negative $x$ - axis, 7 units away from the directrix.