Tangents PA and PB 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 between the tangents satisfies  $\frac{\pi }{3}\le \theta \le \frac{2\pi }{3}$. The co-ordinates of the point P, if the perimeter of  $\Delta PAB$  is maximum are.

 $\angle OAB=\angle APO=\frac{\theta }{2} \therefore AP=BP=r\cot \frac{\theta }{2}andAB=2r\cos \frac{\theta }{2}$
$\therefore$  Perimeter of  $\Delta APB,L=2r(\cot \frac{\theta }{2}+\cos \frac{\theta }{2})$
 
$\frac{dL}{dr}=r(-\cos e{c}^2\frac{\theta }{2}-\sin \frac{\theta }{2})<0$ 
For all  $\theta \in [\frac{\pi }{3},\frac{2\pi }{3}]$
$\therefore$  L is maximum when  $\theta =\frac{\pi }{3}$  and then   $OP=r\cos ec\frac{\theta }{2}=2r$ .
So, the co-ordinates of P are  $(2r\cos {45}^0,2r\sin {45}^0)$ .