91Ó°ÊÓ

Express the given equations in polar coordinates. (a) \(y=-3\) (b) \(x^{2}+y^{2}=5\) (c) \(x^{2}+y^{2}+4 x=0\) (d) \(x^{2}\left(x^{2}+y^{2}\right)=y^{2}\)

Sketch the hyperbola, and label the vertices, foci, and asymptotes. (a) \(\frac{(y+4)^{2}}{3}-\frac{(x-2)^{2}}{5}=1\) (b) \(16(x+1)^{2}-8(y-3)^{2}=16\)

Sketch the polar curve and find polar equations of the tangent lines to the curve at the pole. $$r=2 \cos 3 \theta$$

Find parametric equations for the curve, and check your work by generating the curve with a graphing utility. A circle of radius \(5,\) centered at the origin, oriented clockwise.

Prove that a hyperbola is an equilateral hyperbola if and only if \(e=\sqrt{2}\).

Find parametric equations for the curve, and check your work by generating the curve with a graphing utility. The portion of the circle \(x^{2}+y^{2}=1\) that lies in the third quadrant, oriented counterclockwise.

Sketch the polar curve and find polar equations of the tangent lines to the curve at the pole. $$r=4 \sin \theta$$

Sketch the hyperbola, and label the vertices, foci, and asymptotes. (a) \(x^{2}-4 y^{2}+2 x+8 y-7=0\) (b) \(16 x^{2}-y^{2}-32 x-6 y=57\)

Find parametric equations for the curve, and check your work by generating the curve with a graphing utility. A vertical line intersecting the \(x\) -axis at \(x=2,\) oriented upward.

Sketch the polar curve and find polar equations of the tangent lines to the curve at the pole. $$r=4 \sqrt{\cos 2 \theta}$$