Deck 10: Topics From Analytic Geometry

Find the standard form of the equation of the ellipse with the given characteristics.
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Foci: $( 0,0 ) , ( 8,0 )$ ; major axis of length 10.
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Graph the hyperbola.
​ $9 x ^ { 2 } - 25 y ^ { 2 } = 225$ ​

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.
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Describe the graph of the polar equation and find the corresponding rectangular equation. Select the correct graph.
​ $$r = 5 \sec \theta$$ ​

Identify the conic as a circle or an ellipse then find the center.
​ $$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 25 } = 1$$ ​

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin.
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Passes through the point $\left( - 3 , \frac { 1 } { 4 } \right)$ ; vertical axis
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Select the polar equation of graph.
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A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. If this velocity is multiplied by $\sqrt { 2 }$ , the satellite will have the minimum velocity necessary to escape Earth's gravity and will follow a parabolic path with the center of Earth as the focus. (Hints: The radius of Earth is 4,000 miles.)
​ Find the distance between the surface of the Earth and the satellite when $\theta = 40 ^ { \circ }$ .
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Find a polar equation of the conic with its focus at the pole.
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Conics
Vertex or vertices
Parabola $\left( 5 , - \frac { \pi } { 2 } \right)$ ​

Graph the hyperbola.
​ $$9 y ^ { 2 } - 9 x ^ { 2 } + 72 y + 54 x = 18$$ ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.
​ $x = t + 1$ $y = \frac { t } { t + 1 }$ ​

Find a set of parametric equations for the rectangular equation.
​ $\begin{array} { l } t = 3 - x \\x = 4 y - 3\end{array}$ ​

Identify the conic by writing the equation in standard form.
​ $4 x ^ { 2 } + 9 y ^ { 2 } + 16 x - 90 y + 205 = 0$ ​

Find a polar equation of the conic with its focus at the pole.
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Conics
Eccentricity
Directrix
Hyperbola $e = \frac { 3 } { 2 }$ $x = - 4$ ​