Question

Find the equation of the circle which cuts orthogonally each of the circles

S' ≡ x² + y² + 2x + 17y + 4 = 0

S'' ≡ x² + y² + 2x + 17y + 4 = 0

S''' ≡ x² + y² + 2x + 17y + 4 = 0

Answer 1

Radical axis of S' = 0 and S'' = 0 is S' − S'' ≡ 5x − 11y + 7 = 0 and the radical axis of S'' = 0 and S'' = 0 is S'' − S'''  - 8x − 16y + 8 = 0. That is,

5x - 11x = -7

and  x - 2y = -1

Solving these equations, we obtain the radical centre as (3, 2). If t is the length of the tangent from (3, 2) to the circle S' = 0, we have

Therefore, the required circle is (x − 3)² + (y -2)² = 57