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?
Select the correct option:
Solution
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.
🔒 Solution Hidden from View
Submit your answer to unlock the detailed step-by-step solution.
More ellipse Practice Questions
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.
Looking for more practice? Explore all mathematics questions or browse coordinate geometry questions on RankGuru.