91Ó°ÊÓ

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

Find an equation for the indicated conic section. Ellipse with foci (0,1) and (0,5) and vertices (0,-1) and (0,7)

Find the slope of the tangent line to the polar curve at the given point. $$r=3 \sin \theta \text { at } \theta=0$$

In exercises find the slopes of the tangent lines to the given curves at the indicated points. $$\left\\{\begin{array}{ll} x=\cos 2 t & (\mathrm{a}) t=\frac{\pi}{4},(\mathrm{b}) t=\frac{\pi}{2},(\mathrm{c})\left(\frac{\sqrt{2}}{2}, 1\right) \\ y=\sin 4 t & \end{array}\right.$$

Find an equation for the indicated conic section. Ellipse with foci (1,2) and (1,4) and vertices (1,1) and (1,5)

Plot the given polar points \((r, \theta)\) and find their rectangular representation. $$\left(5,-\frac{\pi}{2}\right)$$

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

Find polar equations for and graph the conic section with focus (0,0) and the given directrix and eccentricity. Directrix \(y=2, e=1.2\)

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

In exercises find the slopes of the tangent lines to the given curves at the indicated points. $$\left\\{\begin{array}{ll} x=\cos 2 t & \text { (a) } t=\frac{\pi}{2}, \text { (b) } t=\frac{3 \pi}{2}, \text { (c) }(1,0) \\ y=\sin 3 t & \end{array}\right.$$