91Ó°ÊÓ

Sketch the graph of \(r=4 \sin \theta\) over each interval. (a) \(0 \leq \theta \leq \frac{\pi}{2}\) (b) \(\frac{\pi}{2} \leq \theta \leq \pi\) (c) \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\)

A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at the focus will be reflected to the other focus. The mirror in the figure has the equation \(\left(x^{2} / 36\right)-\left(y^{2} / 64\right)=1\). At which point on the mirror will light from the point \((0,10)\) be reflected to the other focus?

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. It is possible for a parabola to intersect its directrix.

For a point \(P\) on an ellipse, let \(d\) be the distance from the center of the ellipse to the line tangent to the ellipse at \(P\). Prove that \(\left(P F_{1}\right)\left(P F_{2}\right) d^{2}\) is constant as \(P\) varies on the ellipse, where \(P F_{1}\) and \(P F_{2}\) are the distances from \(P\) to the foci \(F_{1}\) and \(F_{2}\) of the ellipse.