Let tan gents drawn to circle, x^2 + y^2 = 16 from point P(0, h) meet x- axis at point A and B

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10-1.Circle and System of Circles

hard

Let the tan gents drawn to the circle, $x^2 + y^2 = 16$ from the point $P(0, h)$ meet the $x-$ axis at point $A$ and $B.$ If the area of $\Delta APB$ is minimum, then $h$ is equal to

A

$4\sqrt 2$

B

$3\sqrt 3$

C

$3\sqrt 2$

D

$4\sqrt 3$

(JEE MAIN-2015)

Solution

$OP = \frac{4}{{\sin \,\theta }}$

$OB = \frac{4}{{\cos \,\theta }}$

Area $ = OP \times OB = \frac{{16}}{{\sin \,\theta \,\cos \,\theta \,}} = \frac{{32}}{{\sin \,2\theta }}$

least value $\sin \,2\theta = 1;\theta = {45^0}$

So, $h = \frac{4}{{\sin \,{{45}^0}}} = 4\sqrt 2 $

Standard 11

Mathematics

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