The ordinates of the points P and Q on the parabola with focus ( 3 , 0 ) and directrix x = - 3 are in the ratio 3 : 1 . If R ( α , β ) is the point of intersection of the tangents to the parabola at P and Q , then β 2 α is equal to

Step-by-step solution

Given, Focus ( 3 , 0 ) , Directrix, x = - 3 So, the equation of parabola will be, y 2 = 12 x Now taking the parametric point 3 t 2 , 6 t and plotting the diagram we get, Now we know that R ( α , β ) will be the aithematic mean of ordinate and geometric mean of abscissa will be, Hence, β = 6 t 1 + t 2 2 and α = 3 t 1 2 × 3 t 2 2 = 3 t 1 t 2 So, putting the value in β 2 α , ⇒ β 2 α = 9 t 1 + t 2 2 3 t 1 t 2 = 9 4 t 2 2 3 × 3 t 2 2 = 16     ∵ 6 t