Let L be a tangent line to the parabola y 2 = 4 x - 20 at 6 , 2 . If L is also a tangent to the ellipse x 2 2 + y 2 b = 1 , then the value of b is equal to :

Step-by-step solution

The equation of a tangent to a parabola y 2 = 4 a x - h at a point x 1 ,   y 1 on it is y y 1 = 4 a x + x 1 2 - h . Hence, the equation of tangent to parabola y 2 = 4 x - 20 at 6 ,   2 is 2 y = 2 ( x + 6 ) - 20 ⇒ y = x - 4 . Also, a line y = m x + c is a tangent to the ellipse x 2 a 2 + y 2 b 2 = 1 if c 2 = a 2 m 2 + b 2 . Hence, by using the condition of tangency for ellipse x 2 2 + y 2 b = 1 and the line y = x - 4 , we get 16 = 2 ( 1 ) 2 + b ⇒ b = 14 .