Question

Tangents PQ and PR are drawn to the circle S ≡ x² + y² + − a² = 0 from the point P(x₁, y₁) to touch the circle at Q and R. Determine the equation of the circumcircle of  ΔPQR.

Answer 1

QR being the chord of contact of P(x₁, y₁) with respect to the circle S = 0, its equation is S₁  ≡ xx₁ + yy₁ − a² = 0 Any circle passing through Q and R is of the form S + λL = 0 where S ≡ x² + y²  − a² = 0 and L ≡ xx₁ + yy₁ − a² = 0. Therefore

 S + λL ≡ x² + y²  − a² + λ(xx₁+ yy₁ - a²) = 0

which is the circumcircle of ΔPQR, with the condition that it passes through P(x₁, y₁). Therefore

Therefore, the equation of the circumcircle of  ΔPQR is x² + y² − xx₁ − yy₁ = 0.