# Thread: Prove tangent bisects line

1. ## Prove tangent bisects line

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!

2. You have found the co-ordinates of R and T.
Find the mid point R and T. That will be the co-ordinates of Q.