91Ó°ÊÓ

Sketch the graph of the given equation. $$ \frac{(x+3)^{2}}{4}+\frac{(y-2)^{2}}{8}=0 $$

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=6 s^{2}, y=-2 s^{3} ; s \neq 0 $$

In Problems 17-22, find the Cartesian equations of the graphs of the given polar equations. \(r^{2}-6 r \cos \theta-4 r \sin \theta+9=0\)

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=6\)

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$y^{2}=-15 x,(-3,-3 \sqrt{5})$$

Sketch the graph of the given equation. $$ x^{2}+4 y^{2}-2 x+16 y+1=0 $$

Find the slope of the tangent line to each of the following curves at \(\theta=\pi / 3\). (a) \(r=2 \cos \theta\) (b) \(r=1+\sin \theta\) (c) \(r=\sin 2 \theta\) (d) \(r=4-3 \cos \theta\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=2 \theta^{2}, y=\sqrt{5} \theta^{3} ; \theta \neq 0 $$

Find the equations of the tangent and the normal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line. $$x^{2}=4 y,(4,4)$$

Find all points on the cardioid \(r=a(1+\cos \theta)\) where the tangent line is (a) horizontal, and (b) vertical.