Question

The coordinates of the point on the parabola y² = 8x which is at minimum distance from the circle x² + (y + 6)² = 1 are

विकल्प

• (2, – 4)

• (18, – 12)

• (2, 4)

• None of these

Answer 1

(2, – 4)

Explanation:

A point on the parabola is at a minimum distance from the circle if and only if it is at a minimum distance from the centre of the circle.

A point on the parabola y² = 8x is of the type P(2t², 4t).

Centre C of circle x² + (y+ 6)² = 1 is (0, – 6).

∴ CP² = 4t⁴ + (4t + 6)² = 4(t⁴ + 4t² + 12t + 9)

⇒ `d/(dx) (CP)^2 = 4(4t^3 + 8t + 12) = 16(t + 1)(t^2 - t + 3)`

Also `d^2/(dt^2) (CP)^2 = 48t^2 + 32`

`d/(dt^2) (CP^2)` = 0

⇒ t = – 1 ......(Real value)

And `d^2/(dt^2) (CP)^2|_(t = - 1)` = 80 > 0

∴ Required point is (2, – 4).