Ellipse
An ellipse has foci separated by a distance of 6 units and a sum of focal radii equal to 10 for every point; what is its standard equation centred at the origin?
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About This Question
- Subject
- mathematics
- Chapter
- coordinate geometry
- Topic
- ellipse
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
x2/25+y2/16=1
An ellipse is defined by a constant sum of distances to two foci equal to 2a, while the focal separation is 2c, and the semi-minor axis follows from b^2 = a^2 - c^2. The sum of focal radii is 10, so 2a = 10 and a = 5, giving a^2 = 25. The foci are 6 apart, so 2c = 6 and c = 3. Then b^2 = a^2 - c^2 = 25 - 9 = 16. With foci on the x-axis, the standard equation is x^2/25 + y^2/16 = 1. Option x^2/25 + y^2/9 = 1 wrongly sets b^2 = c^2. Option x^2/16 + y^2/25 = 1 places the major axis vertically, contradicting horizontal foci. Option x^2/100 + y^2/64 = 1 doubles the axes by misreading 2a as a. This is the standard JEE Advanced definition-to-equation construction. Plausibility check: eccentricity e = c/a = 3/5 lies strictly between 0 and 1, and b^2 = 16 is positive, confirming a genuine, properly oriented ellipse.
This hard difficulty mathematics question is from the chapter coordinate geometry, covering the topic of ellipse. It appeared in the 2025 exam.
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