Mathematics - Circle Question with Solution | TestHub

Let equation of tangent be$x\cos \mathrm{ }\mathrm{\alpha }+y\sin \mathrm{ }\mathrm{\alpha }=4$

Now, its points of intersection with co-ordinate axis are

$\mathrm{A}\left(\frac{4}{\mathrm{cos\alpha }},0\right),\mathrm{ }\mathrm{P}\left(0,\frac{4}{\mathrm{sin\alpha }}\right)$

Now,$\mathrm{ar}\left(∆\mathrm{PAB}\right)=2\left(\mathrm{ar}∆\mathrm{PAO}\right)$

$=2\times \frac{1}{2}\left(\mathrm{OA}\times \mathrm{OP}\right)$

$=\left|\frac{4}{\sin \mathrm{\alpha }}\mathrm{ }\mathrm{ }.\mathrm{ }\frac{4}{\cos \mathrm{\alpha }}\right|=\left|\frac{32}{\sin 2\mathrm{\alpha }}\right|\ge 32$

which is maximum if,$\mathrm{\alpha }=\frac{\mathrm{\pi }}{4}$

$\therefore \mathrm{ }\mathrm{ }\mathrm{ }\mathrm{P}\left(0,\frac{4}{\sin \frac{\mathrm{\pi }}{4}}\right)$

$\therefore \mathrm{ }\mathrm{ }\mathrm{ }\mathrm{h}=4\sqrt{2}$