Problem

The ellipse $E$ has equation

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \qquad a>b>0$$

The point $P(a\cos\theta, b\sin\theta)$ lies on E where $0<\theta<\frac{\pi}{2}$

(a) Use calculus to show that an equation of the normal to E at P is

$$by = ax\tan\theta + (b^2-a^2)\sin\theta$$

The normal to E at P meets E again on the y-axis at the point B.

Given that O is the origin and that the area of triangle OBP is $\dfrac{3b^2}{4}$

(b) show that $\sin\theta = \dfrac{1}{2}$

(c) determine, in terms of $a$ only, the exact coordinates of the point P.