Centre of a circle which is orthogonal to the circle x^2+y^2=25 and (x-7)^2+(y-5)^2=4 and which has the least radius is

Let the centres of the two circles be O₁ and O₂.

 

Any circle orthogonal to the two circles will have its centre on the radical axis and will pass through the limiting points of the co-axial system formed by the two given circles.

 

Its clear that among such circles, the one formed with the limiting points as the diameter will have the least radius. Hence we have to find the intersection of O₁O₂ and the radical axes. i.e. of the lines

 

5x = 7y and 14x+10y = 95 .