91Ó°ÊÓ

Plot the given polar points \((r, \theta)\) and find their rectangular representation. $$(2, \pi)$$

Find an equation for the indicated conic section. Parabola with focus (1,2) and directrix \(y=-2\)

Sketch the plane curve defined by the given parametric equations and find a corresponding \(x\) -y equation for the curve. $$\left\\{\begin{array}{l}x=1+2 \cos t \\\y=-2+2 \sin t\end{array}\right.$$

Find the slope of the tangent line to the polar curve at the given point. $$r=\sin 3 \theta \text { at } \theta=\frac{\pi}{2}$$

Plot the given polar points \((r, \theta)\) and find their rectangular representation. $$(-2, \pi)$$

Find the are length of the curve; approximate numerically, if needed. $$\left\\{\begin{array}{l} x=t^{3}-4 t \\ y=t^{2}-3 \end{array},-2 \leq t \leq 2\right.$$

Find an equation for the indicated conic section. Parabola with focus (3,0) and directrix \(x=1\)

Sketch the plane curve defined by the given parametric equations and find a corresponding \(x\) -y equation for the curve. $$\left\\{\begin{array}{l}x=-1+2 t \\\y=3 t\end{array}\right.$$

In exercises find the slopes of the tangent lines to the given curves at the indicated points. $$\left\\{\begin{array}{ll} x=2 \cos t & (a) t=\frac{\pi}{4},(b) t=\frac{\pi}{2},(c)(0,3) \\ y=3 \sin t & \end{array}\right.$$

Find polar equations for and graph the conic section with focus (0,0) and the given directrix and eccentricity. Directrix \(x=2, e=1\)