The circles are given as x 2 + y 2 = 12 … (1)

and x 2 + y 2 – 5x + 3y – 2 = 0 … (2)

If A and B are the points of intersection of (1) and (2), clearly AB will be the common chord whose equation will be

(x 2 + y 2 – 12) – (x 2 + y 2 – 5x + 3y – 2) = 0

or 5x – 3y – 10 = 0 … (3)

If p be the point where the tangents at A and B with respect to (1), meet each other, AB will be the chord of contact of P. Let the co-ordinates of P be ( α , β ). Equation of the chord of contact of ( α , β ) with respect to (1) is

x α + y β – 12 = 0 … (4)

As (3) and (4) represent the same equation, comparing the coeffs, we get

α /5 = β /–3 = –12/–10, by which, we get

α = 6 and β = –18/5

Hence the required point is (6, –18/5).