JEE Main 2020 · 5 September · morning · Single correct · +4 / −1

If the co-ordinates of two points A and B
are ( √7 ,0 ) and ( - √7 ,0 ) respectively and
P is any point on the conic, 9x² + 16y² = 144, then PA + PB is equal to :

  1. (a) 8
  2. (b) 9
  3. (c) 16
  4. (d) 6
Answer and solution

Answer (a) 8

Solution

Maths sir · 4 steps

Notice that A and B are the foci of this ellipse, so PA + PB is just the major axis length.

  1. Standard form. Divide 9x² + 16y² = 144 by 144 to get x²/16 + y²/9 = 1. So a² = 16 and b² = 9, which gives a = 4 and b = 3.
  2. Locate the foci. The foci are at (± ae, 0), where ae = √(a² - b²).

    ae = √(16 - 9) = √7

  3. Use the focal property. So the foci are (√7, 0) and (-√7, 0), which are exactly A and B. For any point on an ellipse, the sum of its distances from the two foci equals the major axis length 2a.
  4. So. PA + PB = 2a = 2 × 4 = 8, which is option (a).

The idea to keep. When a question gives you two points on the major axis, check whether they are the foci at (±√(a² - b²), 0). If they are, the sum of distances is 2a for every point on the ellipse.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Ellipse

Asked about 100 times in JEE Main since 2020. This question is from 2020.