Question

Show that the two circles S ≡ x² + y² + 4y − 1 = 0 and S' ≡ x² = y² + 6x + y + 8 = 0 touch each other. Find the common tangent at the point of contact and the point of contact.

Answer 1

A (0, 2) , r₁ = √5, respectively, are the centre and the radius of S = 0. Similarly B = (−3, −1/2) and  r₂ = √5/2, respectively, are the centre and the radius of S' = 0. The distance between the centres (see Fig.) is given by

Therefore, S = 0, S' = 0 touch each other externally. The common tangent is S' − S' ≡ 2x − y + 3 = 0. Suppose P(x, y) is the point of contact. Therefore