Question

Let C₁ be the circle of radius 1 with center at the origin. Let C₂ be the circle of radius r with center at the point A = (4, 1), where 1 < r < 3. Two distinct common tangents PQ and ST of C₁ and C₂ are drawn. The tangent PQ touches C₁ at P and C₂ at Q. The tangent ST touches C₁ at S and C₂ at T. Midpoints of the line segments PQ and ST are joined to form a line which meets the x -axis at a point B. If AB = √5, then the value of r² is  _____.

Answer 1

Let M and N be midpoints of PQ and ST respectively.

⇒ MN is a radical axis of two circles