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Tangents are drawn to the circle x2 + y2 = 12 at the points where it is met by the circle x2 + y2 – 5x + 3y – 2 = 0; find the point of intersection of these tangents.

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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 

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