Question

Let C₁ be the circle of radius 1 with the 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 that meets the x-axis at a point B. If AB = √5, then the value of r² is : 

Answer 1

Consider points M and N as the midpoints of line segments PQ and ST respectively.
As a result, the line MN serves as the radical axis connecting two circles.
Equation are as follows :
C₁ : x² + y² = 1 ……(i)
C₂ : (x - 4)² + (y - 1)² = r²
⇒ x² + y² - 8x - 2y + 17 - r² = 0  …….(ii)
Now, from the equation (i) and (ii), we get :
Equation of MN :
8x + 2y - 18 + r² = 0
As, B is on x-axis
⇒ \(B(\frac{18-r^2}{8},0)\)
So, AB = \(\sqrt5\)
\(⇒\sqrt{(\frac{18-r^2}{8}-4)^2+1}=\sqrt5\)   …… (By distance formed a)
⇒ On solving, we get :
r² = 2