Question

C₁ and C₂ are two concentric circles. The radius of C₂ is twice that of C₁ . From a point P on C₂, tangents PA and PB are drawn to C₁ . Prove that the centriod of ΔPAB lies on C₁.

Answer 1

Let the circle C₁ be x² + y² = a². So the equation of C₂ must be x² + y² = 4a2. Let P(h, k) be on C₂ so that

h² + k² = 4a²  ...(1)

Equation of AB is

hx + ky = a²     ............(2)

Substituting y = (a² – hx)/k in x² + y² = a², we get

Therefore, 4x² - 2hx + a² - k² = 0 has two distinct roots, say x₁ and x₂. Hence

Suppose G(x, y) is the centroid of  ΔPAB (see Fig). In such case

Hence, the centroid G( , x y C )lies on C₁.