If the common tangents of x 2 + y 2 = r 2 and x 2 16 + y 2 9 = 1 form a square, then the area (in sq. units) of the square is

Step-by-step solution

Let, the equation of tangent in slope form for the circle be y = m x ± r 1 + m 2 and the equation of tangent in slope form for the ellipse be y = m x ± 16 m 2 + 9 Now, for common tangets, y = m x ± r 1 + m 2 is same as y = m x ± 16 m 2 + 9 ⇒ r 2 + r 2 m 2 = 16 m 2 + 9 ⇒ r 2 - 16 m 2 + r 2 - 9 = 0 Since, here m 1 m 2 = - 1 ⇒ 2 r 2 = 25 ⇒ r = 5 2 So, the side of the square = 2 r = 5 2 ⇒ area of the square is 50 sq. units