Question

Show that the circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 touch each other. Also find the point of contact and common tangent at this point of circle.

Answer 1

Equations of the circles are

x2 + y2 – 4x – 6y – 12 = 0

and x2 + y2 + 6x + 18y + 26 = 0

Centres are C1(2, 3), C2 = (–3, –9)

∴ Circle touch externally

Equation of common tangent is S1 – S2 = 0

–10x – 24y – 38 = 0

5x + 12y + 19 = 0

The point of contact P divides C1C2 in the ratio 5 : 8

Co–ordinates of P are

The common tangent at this point of contact is

i.e., 5x + 12y + 19 = 0