Let a line L : 2 x + y = k , k > 0 be a tangent to the hyperbola x 2 - y 2 = 3 . If L is also a tangent to the parabola y 2 = α x , then α is equal to:

Step-by-step solution

A line y = m x + c is a tangent to the hyperbola x 2 a 2 - y 2 b 2 = 1 if c 2 = a 2 m 2 - b 2 . Given that, tangent to hyperbola x 2 3 - y 2 3 = 1 is 2 x + y = k or y = - 2 x + k Thus, we have, slope m = - 2 ,   c =   k   &   a 2 = b 2 = 3 k 2 = 3 - 2 2 - 3 ⇒   k 2 = 9 Given k > 0 ,   ⇒   k = 3 . Thus, the equation of the tangent to the hyperbola is y = - 2 x + 3 . Given this line is also a tangent to the parabola, y 2 = α x and a line y = m x + c is tange