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