Question

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

• A. (6, –18/5)
• B. (6, 18/5)
• C. (18/5, 6)
• D. None

Answer 1

Answer: (A)

(6, –18/5)

Answer 2

The circles are given as x² + y² = 12 … (1)

and x² + y² – 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

(x² + y² – 12) – (x² + y² – 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 (a, b). Equation of the chord of contact of (a, b) with respect to (1) is

xa + yb – 12 = 0 … (4)

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

a/5 = b/–3 = –12/–10, by which, we get

a = 6 and b = –18/5

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