Question

Suppose two circles pass through the points (0, a) and (0, −a) and touch the line y = mx + c. If the two circles cut orthogonally, then show that c2 = a2(2 + m2).

Answer 1

The centre of a circle passing through the points (0, −a) and (0, a) must lie on x-axis. Let the equation of any such circle be

• ≡ x2 + y2 + 2gx - a2 = 0

• = 0 touches the line y = mx + c implies that

Let g1 and g2 be two the roots of this quadratic equation. Hence

g1 + g2 = -2cm

g1g2 = a2 (a + m2) - c2

Now, the circles x2 + y2 + 2g1x − a2 and x2 + y2 - 2g2x − a2 = 0 cut orthogonally implies that