91Ó°ÊÓ

Find the area of the region that lics inside the first curve and outside the second curve. $$r=2 \cos \theta, \quad r=1$$

Show that a conic with focus at the origin, eccentricity \(e\) and directrix \(x=-d\) has polar equation $$r=\frac{e d}{1-e \cos \theta}$$

Find a polar equation for the curve represented by the given Cartesian equation. $$x^{2}+y^{2}=2 c x$$

Find the area of the region that lics inside the first curve and outside the second curve. $$r=1-\sin \theta, \quad r=1$$

Graph the curve in a viewing rectangle that displays all the important aspects of the curve. $$x=t^{4}+4 t^{3}-8 t^{2}, \quad y=2 t^{2}-t$$

Find a polar equation for the curve represented by the given Cartesian equation. $$x y=4$$

Show that a conic with focus at the origin, eccentricity \(e\) and directrix \(y=d\) has polar equation $$r=\frac{e d}{1+e \sin \theta}$$

For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. (a) A line through the origin that makes an angle of \(\pi / 6\) with the positive \(x\) -axis (b) A vertical line through the point \((3,3)\)

Show that a conic with focus at the origin, eccentricity \(e\) and directrix \(y=-d\) has polar equation $$r=\frac{e d}{1-e \sin \theta}$$

Show that the curve \(x=\cos t, y=\sin t \cos t\) has two tangents at \((0,0)\) and find their equations. Sketch the curve.