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Exam 11: Conic Sections

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Question 51

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Determine the equation of the conic $x ^ { 2 } - 2 x = y ^ { 2 }$ in $X Y$ - coordinates if the axes are rotated through an angle $\phi = 45 ^ { \circ }$ .

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Question 52

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Determine the equation of the conic $x y = x ^ { 2 } - y ^ { 2 }$ in $X Y$ - coordinates if the axes are rotated through an angle $\phi = 60 ^ { \circ }$ .

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Question 53

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Find the vertices, foci, and eccentricity of the ellipse given by $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 36 } = 1$ . Determine the lengths of the major and minor axes, and sketch the graph.

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

Find an equation of the parabola whose graph is shown.

Find an equation of the parabola whose graph is shown.

Write a polar equation of a parabola with vertex at $( 2,3 \pi / 2 )$ .

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

Find an equation for a parabola with vertex at the origin, and with directrix $y = - \frac { 1 } { 2 }$ .

Write a polar equation of a hyperbola with eccentricity 3 and directrix $r = 2 \csc \theta$ .

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

Complete the square to determine whether the equation $y ^ { 2 } + 25 = 14 y - 16 x$ 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.

Find an equation for the hyperbola whose graph is shown.

Complete the square to determine whether the equation $\frac { 5 } { 2 } y ^ { 2 } + 5 y - \frac { 5 } { 4 } x ^ { 2 } + 5 x = \frac { 7 } { 2 }$ 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.

Use the discriminant to determine if the graph of the equation $\frac { 3 } { 4 } x ^ { 2 } + \frac { \sqrt { 3 } } { 2 } x y + \frac { 1 } { 4 } y ^ { 2 } + 4 x - 4 \sqrt { 3 } y = 0$ is a parabola, an ellipse or a hyperbola, and then use a rotation of axes to eliminate the $x y$ - term, and sketch the graph.

Find an equation for the ellipse whose graph is shown.

Find an equation for a parabola with vertex at the origin, and with directrix $x = - \frac { 2 } { 3 }$ .

Complete the square to determine whether the equation $64 x ^ { 2 } + y ^ { 2 } + 128 x - 10 y + 73 = 0$ 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 the focus, directrix, and focal diameter of the parabola $x ^ { 2 } = 7 y$ , and sketch its graph.