If the chord joining points $P\left(\alpha \right)andQ\left(\beta \right)$ on the ellipse $\left(\frac{x^2}{a^2}\right)+\left(\frac{y^2}{b^2}\right)=1$ subtends a right angle at the vertex $A\left(a,0\right),$ then prove that $\tan \left(\frac{a}{2}\right)\tan \left(\frac{\beta }{2}\right)=-\frac{b^2}{a^2}\dot{}$

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