A circle of radius 2 unit passes through the vertex and the focus of the parabola y 2 = 2 x and touches the parabola y = x - 1 4 2 + α , where α > 0 . Then 4 α - 8 2 is equal to ______.

Step-by-step solution

The vertex and focus of parabola y 2 = 2 x are V 0 , 0 and S 1 2 , 0 respectively Let the equation of circle passing through vertex and focus of the parabola be x - h 2 + y - k 2 = 4 ∵ Circle passes through 0 , 0 ⇒ h 2 + k 2 = 4           ⋯ 1 ∵ Circle passes through 1 2 , 0 1 2 - h 2 + k 2 = 4 ⇒ h 2 + k 2 = h + 15 4           ⋯ 2 On solving 1 and 2 4 = h + 15 4 ⇒ h = 1 4 ⇒ k = 63 4 here k = - 63 4 is rejected as circle