Let the length of the latus rectum of an ellipse with its major axis along -axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: B
Solution
For an ellipse x²/a² + y²/b² = 1, latus rectum = 2b²/a = 8, so b² = 4a.
Condition: distance between foci = minor axis length → 2c = 2b → c = b.
Since b² = a² - c² = a² - b², we get 2b² = a², so a = b√2.
From b² = 4a = 4b√2: b = 4√2, a = b√2 = 8.
So a² = 64, b² = 32.
Check option 2: (4√3, 2√2) → (48/64) + (8/32) = 3/4 + 1/4 = 1 ✓
Condition: distance between foci = minor axis length → 2c = 2b → c = b.
Since b² = a² - c² = a² - b², we get 2b² = a², so a = b√2.
From b² = 4a = 4b√2: b = 4√2, a = b√2 = 8.
So a² = 64, b² = 32.
Check option 2: (4√3, 2√2) → (48/64) + (8/32) = 3/4 + 1/4 = 1 ✓
[JEE Main 2019 · 11 Jan · Shift II]