Exam 8: Polar Coordinates and Parametric Equations

Write $3 - 3 \sqrt { 3 } i$ in polar form, with $\theta$ between $0$ and $2 \pi$ .

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 16$

Write $- 25 i$ in polar form with argument $\theta$ between $0$ and $2 \pi$ .

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( 0 , - \sqrt { 5 } )$

Find the rectangular coordinates of the point with polar coordinates $( \sqrt { 3 } , \pi / 6 )$ .

Convert the equation $x y = 1$ to polar form.

Sketch the curve represented by $x = t - 1$ , $y = \left| t ^ { 2 } - 1 \right|$ and find a rectangular-coordinate equation for the curve.

Find parametric equations for the line passing through the points $( 3,5 )$ and $( - 5,7 )$ .

Solve $z ^ { 4 } - 10,000 = 0$ .

Find parametric equations for the line of slope $- \frac { 4 } { 5 }$ passing through the point $( 7 , - 1 )$ .

Convert the polar equation to rectangular coordinates. $r+ \cos \theta = 4$

Use a graphing device to draw the curve represented by , , $0 \leq t \leq 2 \pi$ .

Find parametric equations for the parabola $y = x ^ { 2 } - 3$ .

Sketch the graph of the polar equation. $\theta = \pi / 4$

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $\left( x ^ { 2 } + y ^ { 2 } + 3 y \right) ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right)$

Sketch the curve represented by $x = \sec t$ , $y = \cot t$ and find a rectangular-coordinate equation for the curve.

Write $- 1 - i$ in polar form, with $\theta$ between $0$ and $2 \pi$ .

Sketch the graph of the polar equation. $r = \cos \theta$

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 25$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r ^ { 2 } = 9 \cos 2 \theta$