If the tangent to the parabola y 2 = x at a point α , β , β > 0 is also a tangent to the ellipse, x 2 + 2 y 2 = 1 then α is equal to:

Step-by-step solution

Since α ,   β lies on the parabola ⇒ α = β 2 ⇒ α ,   β = β 2 ,   β Tangent to a parabola y 2 = 4 a x at a point x 1 ,   y 1 is y y 1 = 2 a x + x 1 Therefore, tangent to the parabola y 2 = x at β 2 ,   β is y ⋅ β = 1 2 x + β 2 ⇒ y = x 2 β + β 2       . . . i Tangent to parabola is also a tangent to ellipse x 2 + 2 y 2 = 1   ⇒ x 2 + y 2 1 2 = 1 And, a line y = m x + c is a t