Tangents..

locus of P is x²+y² = 4

this can easily obtained by observing the nature of the two circles graphically

Reply..some1..

Let (h,k) be the point of the locus at some instant.
We have
k=mh\pm \sqrt{1+m^2}
&
k=Mh\pm \sqrt{3+3M^2}
Use the fact mM=-1, and get the desired locus....now try!

yes, i think i will proceed in the same way as soumik has done...that solves the sum nicely!

No yaar it doesnt help...When we square to get m things complicate a lot..

See both the circles are centred at (0,0)

If you rotate that rectangle ... u will get the locus of P

obviously this is a circle.. which dist. from the centre is 2

So the locus equation is
x²+y²=4

This is for concentric circles only.

For any other system ,, u have to do what soumik has said

Asish has got the best solution...Thanks dear..