Polar Functions

STEP 2006, Math III, #6: Show that in polar coordinates the gradient of any curve at the point is

A mirror is designed so that if an incident ray of light is parallel to a fixed line

the reflected ray passes through a fixed point on . Prove that the mirror intersects any plane containing

in a parabola. You should assume that the angle between the incident ray and the normal to the mirror is the same as the angle between the reflected ray and the normal.

for , and describe the curves , where is an integer.

Show that the area enclosed by such a curve is independent of

for .