91Ó°ÊÓ

Find the polar equation of an equilateral hyperbola with a focus at the pole and vertex \((5,0)\).

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

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.

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

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

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

Let an \(x^{\prime} y^{\prime}\) -coordinate system be obtained by rotating an \(x y\) -coordinate system through an angle \(\theta\). Prove: For every value of \(\theta\), the equation \(x^{2}+y^{2}=r^{2}\) becomes the equation \(x^{\prime 2}+y^{\prime 2}=r^{2}\). Give a geometric explanation.

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

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.

Find an equation for the parabola that satisfies the given conditions. (a) Vertex \((0,0) ;\) focus \((3,0)\). (b) Vertex \((0,0) ;\) directrix \(y=\frac{1}{4}\),