I was set the following problem as part of a homework assignment:

P is a point on the ellipse with parametric equations x=5cost, y=3sint for 0 <= t < 2pi, and F and G are the points (-4,0) and (4,0) respectively. Prove that:

a) FP = 5 + 4cost

b) FP + PG = 10

Let the normal at P make angles θ and Φ with FP and GP respectively. Prove that:

c) tan θ = (4sint)/3

d) θ = Φ

Whilst I found parts (a) and (b) trivial to prove, I could not work out a way to find the tangent of θ for part (c). After taking the problem home, my maths teacher could also not solve the problem! Does anybody have any suggestions?

Thanks in advance.