Tangents are drawn from external point $P(6,8)$ to the circle $x^2+y^2=r^2$. The radius r of circle such that area of triangle formed by the tangents and chord of contact is maximum is ________________

Area of $\Delta PAB=\frac{1}{2}\left(10\cos \theta \right)\left(10\cos \theta \right)\sin \left(2\theta \right)$

$$\begin{gathered} {10}^2{\cos }^3\theta \sin \theta \\ l={\sin }^2\theta +{\cos }^2\theta ={\sin }^2\theta +\frac{{\cos }^2\theta }{3}+\frac{{\cos }^2\theta }{3}+\frac{{\cos }^2\theta }{3}\ge 4{(\frac{{\cos }^6}{3^3}.{\sin }^2\theta )}^{\frac{1}{4}} \\ {\cos }^3\theta .\sin \theta \le \frac{3\sqrt{3}}{16} \\ 100{\cos }^3\theta .\sin \theta =100.\frac{3\sqrt{3}}{16} \\ {\cos }^3\theta .\sin \theta =\frac{3\sqrt{3}}{16}={\left(\frac{\sqrt{3}}{2}\right)}^3.\frac{1}{2} \\ \sin \theta =\frac{1}{2},\cos \theta =\frac{\sqrt{3}}{2} \\ Tan\theta =\frac{1}{\sqrt{3}} \\ Tan\theta \Delta APC \\ Tan\theta =\frac{r}{10\cos \theta } \end{gathered}$$