Let the tangents drawn to the circle, x 2 + y 2 = 16 from the point   P 0 , h meet the x -axis at…

1. $4\sqrt{2}$
2. $3\sqrt{2}$
3. $4\sqrt{3}$
4. $3\sqrt{3}$

Solution

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, $\mathrm{\alpha }=\frac{\mathrm{\pi }}{4}$ 

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

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

Asked in: JEE Main 2015 (10 Apr Online)

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