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A point P on the parabola y 2 = 4 x , the foot of the perpendicular from it upon the directrix and the focus are the vertices of an equilateral triangle. If the area of the equilateral triangle is β sq. units, then the value of β 2 is

Correct answer

48

Step-by-step solution

From the diagram, P M = 1 + t 2 P S = t 2 - 1 2 + 4 t 2 = t 2 + 1 M S = 4 + 4 t 2 = 2 1 + t 2 ⇒ 2 1 + t 2 = 1 + t 2 ∴ P M = 1 + t 2 = 4 ⇒ t 2 = 3 ∴ Area of ∆ P M S = β = 3 4 4 2 = 4 3 sq. units Hence, β 2 = 48

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