Find the focal distance of a point on the parabola y2 = 16x whose ordi

1.	Find the focal distance of a point on the parabola y2 = 16x whose ordinate is 2 times the abscissa.
Answer» Given the equation of the parabola is y² = 16x. Comparing this equation with y² = 4ax, we get ⇒ 4a = 16 ⇒ a = 4 Since ordinate is 2 times the abscissa, y = 2x Substituting y = 2x in y² = 16x, we get ⇒ (2x)² = 16x ⇒ 4x² = 16x ⇒ 4x² – 16x = 0 ⇒ 4x(x – 4) = 0 ⇒ x = 0 or x = 4 When x = 4, focal distance = x + a = 4 + 4 = 8 When x = 0, focal distance = a = 4 ∴ Focal distance is 4 or 8.

Find the focal distance of a point on the parabola y2 = 16x whose ordinate is 2 times the abscissa.

Answer»

Given the equation of the parabola is y² = 16x.

Comparing this equation with y² = 4ax, we get

⇒ 4a = 16

⇒ a = 4

Since ordinate is 2 times the abscissa, y = 2x

Substituting y = 2x in y² = 16x, we get

⇒ (2x)² = 16x

⇒ 4x² = 16x

⇒ 4x² – 16x = 0

⇒ 4x(x – 4) = 0

⇒ x = 0 or x = 4

When x = 4,

focal distance = x + a = 4 + 4 = 8

When x = 0,

focal distance = a = 4

∴ Focal distance is 4 or 8.