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Tangents $P A$ and $P B$ are drawn to the circle $x^{2} + y^{2} = 4$ from an external point $P$ lying on the line $y = x$ such that the angle $θ$ between the tangents satisfies $\frac{π}{3} \leq θ \leq \frac{2 π}{3} .$ The co-ordinates of the point $P,$ if the perimeter of $Δ P A B$ is maximum are

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$(2 \sqrt{2}, 2 \sqrt{2})$

$(\sqrt{2}, \sqrt{2})$

$(\frac{2 \sqrt{2}}{\sqrt{3}}, \frac{2 \sqrt{2}}{\sqrt{3}})$

$(2, 2)$

answer is A.

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Detailed Solution

$∠ O A B = ∠ A P O = \frac{θ}{2}$

$∴ A P = B P = r \cot \frac{θ}{2} and A B = 2 r \cos \frac{θ}{2} .$

$∴ Perimeter of Δ A P B, L = 2 r (\cot \frac{θ}{2} + \cos \frac{θ}{2})$

$\frac{d L}{d r} = r (- \cos e c^{2} \frac{θ}{2} - \sin \frac{θ}{2}) < 0 f o r a l l θ \in [\frac{π}{3}, \frac{2 π}{3}] .$

$∴$ L is maximum when $θ = \frac{π}{3}$ and then $O P = r \cos e c \frac{θ}{2} = 2 r .$

So, the co-ordinates of P are $(2 r \cos 45 °, 2 r \sin 45 °) .$

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