Mid-point of chord on ellipse

Prove that the equation of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $\displaystyle P_1(x_1,y_1)$ is $\displaystyle \frac{xx_1}{a^2}+\frac{yy_1}{b^2}=1$. The tangent at $\displaystyle P_1$ meets the tangent at $\displaystyle P_2(x_2,y_2)$ at T. Show that the line

$\displaystyle \frac{xx_1}{a^2}+\frac{yy_1}{b^2}=\frac{xx_2}{a^2} +\frac{yy_2}{b^2}$

passes through T and through the midpoint of $\displaystyle P_1P_2$. Prove that if $\displaystyle P_1TP_2$ is a right angle, then $\displaystyle \frac{x_1x_2}{a^4}+\frac{y_1y_2}{b^4}=0$

I have proven the first part. I got the equation of the tangent at $\displaystyle P_2$ in the same form as the first. Then I made y the subject of both tangents and found x. did it in the same way to find y.

I found the the line given passed through T, but could not prove that the line passed through the midpoint of $\displaystyle P_1P_2$, $\displaystyle (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$.

Thanks!