Given circles are S1 ≡ x2 + y2 – 12 = 0 ....... (i)
and S2 = x2 + y2 – 5x + 3y – 2 = 0 ....... (ii)
Now equation of common chord of circle (i) and (ii) is
S1 – S2 = 0
i.e. 5x – 3y – 10 = 0 ....... (iii)
Let this line meet circle (i) [or (ii)] at A and B
Let the tangents to circle (i) at A and B meet at P(α, β), then AB will be the chord of contact of the tangents to the circle (i) from P, therefore equation of AB will be
xα + yβ – 12 = 0 ....... (iv)
Now lines (iii) and (iv) are same, therefore, equations (iii) and (iv) are identical