If a>2b>0  then the positive value of m  for which y=mx−b1+m2  is a common tangent to x2+y2=b2 and (x−a)2+y2=b2 is

y=mx−b1+m2 is a tangent to the circle x2+y2=b2 for all values of m. If it also touches the circle (x−a)2+y2=b2 , then the length of the perpendicular from its centre (a, 0)  on this line is equal to the radius b of the circle, which gives ma−b1+m21+m2=±b Taking negative value on R.H.S. We get m=0, so we neglect it.Taking the positive value on R.H.S. we getma=2b1+m2 ⇒ m2(a2−4b2)=4b2 ⇒   m=2ba2−4b2