Question

Find the coordinates of the point on the parabola y² = 8x whose focal distance is 10. 

Answer 1

 Equation of the parabola is y² = 8x 

4a = 8 ⇒ a = 2 

 ⇒S = (2, 0) 

Let P(x, y) be  a point on the parabola

Given SP = 10 

II Method: Given SP = 10 ⇒ SP² = 100 

(x – 2)² + y² = 100 

But y² = 8x 

⇒ (x – 2)² + 8x = 100

⇒ x² – 4x + 4 + 8x – 100 = 0 

⇒ x² + 4x – 96 = 0 ⇒ (x + 12)(x – 8) = 0 

x + 12 = 0 ⇒ x = –12 

x – 8 = 0 ⇒ x = 8 

Case I: x = 8 

y² = 8x = 8 × 8 = 64 

y = ±8  

Coordinates of the required points are (8, 8) and (8, –8) 

Case II: x = –12 

y² = 8 × –12 = –96 < 0 

y is not real.