If two tangents drawn from a point α , β lying on the ellipse 25 x 2 + 4 y 2 = 1 to the parabola y 2 = 4 x are such that the slope of one tangent is four times the other, then the value of 10 α + 5 2 + 16 β 2 + 50 2 equals ______

Step-by-step solution

Let α = 1 5 cos θ ,   β = 1 2 sin θ Equation of tangent to y 2 = 4 x with slope m will be y = m x + 1 m Since the tangent passes through α , β , so 1 2 sin θ = m 5 cos θ + 1 m ⇒ m 2 cos θ 5 - m 1 2 sin θ + 1 = 0 As it is a quadratic equation in m so it will have two roots m 1 and m 2 Given m 1 = 4 m 2 Now, m 1 + m 2 = 1 2 sin θ cos θ 5 m 1 m 2 = 5 cos θ After eliminating m 1 and m 2 , we get, cos θ = - 5 ± 29 2 ⇒ cos 2 &#952