Find the co

Given that we need to find the coordinates of points on the parabola y² = 8x whose focal distance is 4.

We know that the focal distance is the distance from the focus to any point on the parabola.

Comparing the given parabola with standard parabola y² = 4ax. We get,

⇒ 4a = 8

⇒ a = 2

⇒ focus = (a, 0) = (2, 0)

We know that point on y² = 4ax is represented by (at², 2at), where t is any real number.

The point on y² = 8x is (2t², 4t)

We know that the distance between two points (x₁, y₁) and (x₂, y₂) is .

⇒

⇒ 16 = 4 + 4t⁴ - 8t² + 16t²

⇒ 16 = 4t⁴ + 8t² + 4

⇒ 4 = t⁴ + 2t² + 1

⇒ 4 = (t² + 1)²

⇒ t² + 1 = 2

⇒ t² = 1

⇒ t = ±1

The points on parabola is,

⇒ (2t², 4t) = (2(±1)², 4(±1))

⇒ (2t², 4t) = (2, ±4)

∴The points on parabola is (2, ±4).