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NDA 1 2024 Maths, question 64

Topic: Analytical geometry

Q64

In the parabola y2=8xy^{2}=8x, the focal distance of a point PP lying on it is 8 units8\text{ units}. Which of the following statements is/are correct ?

  1. The coordinates of PP can be (6,43)(6,4\sqrt{3}).

  2. The perpendicular distance of PP from the directrix of parabola is 8 units8\text{ units}.

Select the correct answer using the code given below :

  1. (a) 1 only
  2. (b) 2 only
  3. (c) Both 1 and 2Correct answer
  4. (d) Neither 1 nor 2

Solution.

Answer: (c) Both 1 and 2

  1. The parabola has focus (2, 0) and directrix x = −2.
  2. By the focus–directrix property, a focal distance of 8 also gives distance 8 from the directrix, proving statement 2.
  3. Thus x + 2 = 8, so x = 6 and y = ±4√3.
  4. The positive-y point in statement 1 is possible.
  5. Both statements are correct.