Exam 10: Parametric Equations and Polar Coordinates

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: $( \pm 4,0 )$ , asymptotes: $y = \pm \frac { 7 } { 4 } x$

Find the point(s) on the curve where the tangent is horizontal. $x = t ^ { 3 } - 3 t + 2 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 2$

The planet Mercury travels in an elliptical orbit with eccentricity 0.703. Its minimum distance from the Sun is $8 \times 10 ^ { 7 } \mathrm {~km}$ . If the perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ , find the maximum distance (in $\mathrm { km }$ ) from Mercury to the Sun. Select the correct answer.

Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $\frac { 6 } { 7 }$ and directrix $x = - 13$ .

Find parametric equations to represent the line segment from $( - 3,4 )$ to $( 12 , - 8 )$ . Select the correct answer.

Find the area of the region that lies inside both curves. $r = 8 + 2 \sin \theta , r = 7$

If a projectile is fired with an initial velocity of $v _ { 0 }$ meters per second at an angle $\alpha$ above the horizontal and air resistance is assumed to be negligible, then its position after $t$ seconds is given by the parametric equations $x = \left( v _ { 0 } \cos \alpha \right) t , y = \left( v _ { 0 } \sin \alpha \right) t - \frac { 1 } { 2 } g t ^ { 2 }$ where $g$ is the acceleration of gravity $\left( 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$ . If a gun is fired with $\alpha = 55 ^ { \circ }$ and $v _ { 0 } = 440 \mathrm {~m} / \mathrm { s }$ when will the bullet hit the ground?

Graph of the following curve is given. Find its length. Select the correct answer. $r = 6 \cos ^ { 2 } \left( \frac { \theta } { 2 } \right)$

Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $\frac { 6 } { 7 }$ and directrix $x = - 13$ .

Find parametric equations to represent the line segment from $( - 3,4 )$ to $( 12 , - 8 )$ .

The planet Mercury travels in an elliptical orbit with eccentricity $0.703$ . Its minimum distance from the Sun is $8 \times 10 ^ { 7 } \mathrm {~km}$ . If the perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ , find the maximum distance (in km) from Mercury to the Sun.

Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $\frac { 6 } { 7 }$ and directrix $x = - 13$ .

Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$ , length of major axis 8

Find the point(s) on the curve where the tangent is horizontal. Select the correct answer. $x = t ^ { 3 } - 3 t + 2 , y = t ^ { 3 } - 3 t ^ { 2 } + 2$

Find the point(s) on the curve where the tangent is horizontal. $x = t ^ { 3 } - 3 t + 2 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 2$

Match the equation with the correct graph. $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$

Find parametric equations to represent the line segment from $( - 3,4 )$ to $( 12 , - 8 )$ . Select the correct answer.

Find parametric equations for the path of a particle that moves once clockwise along the circle $x ^ { 2 } + ( y - 7 ) ^ { 2 } = 4$ , starting at $( 2,7 )$ .

Find the surface area generated by rotating the lemniscate $r ^ { 2 } = 10 \cos 2 \theta$ about the line $\theta = \pi$ .

Find an equation for the conic that satisfies the given conditions. Select the correct answer. ellipse, foci $( \pm 1,6 )$ , length of major axis 8