P is a point on an ellipse whose major axis is AB the tangent at P meets the minor axis at Q. PA and PB cut the minor axis at R and T. Prove that Q bisects RT.

I put $\displaystyle P(a\cos\theta,b\sin\theta)$ A=(a,0) and B=(-a,0)

equation of tangent at P

$\displaystyle bx\cos\theta+ay\sin\theta=ab$

Point where tangent cuts minor axis, when x=o

$\displaystyle y=\frac{b}{\sin\theta}{$

$\displaystyle Q(0,\frac{b}{\sin\theta})$

equation AP

$\displaystyle y-0=\frac{b\sin\theta}{a\cos\theta-a}(x-a)$

when x=0 $\displaystyle y=-\frac{b\sin\theta}{\cos\theta-1}$

$\displaystyle R(0,-\frac{b\sin\theta}{\cos\theta-1})$

equation BP

$\displaystyle y-0=\frac{b\sin\theta}{a\cos\theta+a}(x+a)$

when x=0, $\displaystyle y=\frac{b\sin\theta}{\cos\theta+1}$

$\displaystyle T(0,\frac{b\sin\theta}{\cos\theta+1})$

$\displaystyle RT=-\frac{b\sin\theta}{\cos\theta-1}-\frac{b\sin\theta}{\cos\theta+1}$

$\displaystyle =-\frac{2b\sin\theta\cos\theta}{\cos^2\theta-1}$

$\displaystyle RQ=-\frac{b\sin\theta}{\cos\theta-1}-\frac{b}{\sin\theta}$

$\displaystyle =-\frac{b(2+\cos\theta)\sin\theta}{\cos^2\theta-1}$

Now I don't know what to do. I can't show that RT=2RQ

Thanks!