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The point on the parabola y2 = 12x, which has a focal distance equal to 12, is:
1. (9, 6√3)
2. (6, 9√3)
3. (-9, 6√3)
4. None of these.

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Correct Answer - Option 1 : (9, 6√3)

Concept:

A Parabola with equation y2 = 4ax:

  • The focus is at the point (a, 0).
  • The general form of any point on the parabola is (at2, 2at).
  • The focal distance of a point is the distance between the point and the focus.

 

Calculation:

The focus of the parabola y2 = 12x = 4(3)x will be at F(3, 0).

Let's say that the required point on the parabola is P(3t2, 6t).

Now, using the distance formula, we have:

PF2 = (3t2 - 3)2 + (6t - 0)2

⇒ 122 = 9(t2 - 1)2 + 36t2

⇒ 122 = 9(t2 + 1)2

⇒ t = ±√3

So, the required points on the parabola are (9, 6√3) and (9, -6√3).

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