Let m 1 and m 2 be the slopes of the tangents drawn from the point P 4 , 1 to the hyperbola H : y 2 25 - x 2 16 = 1 If Q is the point from which the tangents drawn to H have slopes m 1 and m 2 and they make positive intercepts α and β on the x - axis, then ( P Q ) 2 α β is equal to _ _ _ _ _ _ _ .

Step-by-step solution

Given that H : y 2 25 - x 2 16 = 1 Equation of tangent to the hyperbola is : y = m x + 25 - 16 m 2 Passes through 4 , 1 ⇒ y = m x + 25 - 16 m 2 ⇒ 1 = 4 m + 25 - 16 m 2 ⇒ 4 m 2 - m - 3 ⇒ m 1 = 1 , m 2 = - 3 4 ⇒ m 1 = 1 ,    m 2 = 3 4 Equation of tangents with slopes 1 & 3 4 ⇒ y = x - 3 . . . . . . . i ⇒ y = 3 4 x - 4 . . . . . . ii i & ii ⇒ Q ≡ ( - 4 , - 7 ) P ≡ ( 4 , 1 ) ⇒ ( P Q ) 2 = 8 2 + 8 2 = 128 On converting i and i i in intercept for