Question

If Q is the foot of the perpendicular from a point p on the parabola y² = 8(x – 3) to its directrix. S is the focus of the parabola and if SPQ is an equilateral triangle, then find the length of the side of the triangle. 

Answer 1

 Given equation of parabola is (y – 0)² = 8(x – 3) 

Which is of the form (y – k)² = 4a(x – h) 

Where 4a = 8 ⇒ a = 2 

∴ Vertex = (h, k) = (3, 0) 

and focus = (h+a, k) = (3+2, 0) = (5, 0)

Since PQS is an equilateral triangle. 

∠SQP = 60° ⇒ ∠SQZ = 30° 

Also in ∆ SZQ, we have sin 30° = SZ /SQ

∴ SQ (SZ)/sin 30°=2(SZ) 2(4) 8

Hence length of each side of the triangle is 8.