Let the line L : 2 x + y = α pass through the point of the intersection P (in the first quadrant)of the circle x 2 + y 2 = 3 and the parabola x 2 = 2 y . Let the line L touch two circles C 1 and C 2 of equal radius 2 3 . If the centres Q 1 and Q 2 of the circles C 1 and C 2 lie on the y - axis, then the square of the area of the triangle P Q 1 Q 2 is equal to _________.

Step-by-step solution

Given: x 2 + y 2 = 3 and x 2 = 2 y ⇒ y 2 + 2 y − 3 = 0 ⇒ y + 3 y − 1 = 0 ⇒ y = 1 , - 3 Rejected as it gives imaginary value of x ⇒ y = 1 ⇒ x = 2 ⇒ P ≡ 2 , 1 Now, P lies on the line 2 x + y = α . ⇒ 2 2 + 1 = α ⇒ α = 3 For circle C 1 , Q 1 lies on y - axis. Let Q 1 ≡ 0 , β and R 1 = 2 3 (given) Line L act as tangent so applying the condition of tangency ⇒ P = r ⇒ β − 3 3 = 2 3 ⇒ β − 3 = 6 ⇒ β − 3 = 6 , - 6 ⇒ β = 9 , - 3 So, Q 1 ≡ 0 , 9 & Q 2 ≡ 0 , - 3 ⇒ A r Δ P Q 1 Q 2 = 1 2 2 0 0 1 9 − 3 1 1 1 ⇒ A r Δ P Q 1 Q 2 = 1