Ellipse: foci and eccentricity: Class 11 practice
For x²/a² + y²/b² = 1 with a > b, c² = a² − b², foci (±c, 0) and eccentricity e = c/a (between 0 and 1). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Squares and roots
The idea behind ellipse: foci and eccentricity
For x²/a² + y²/b² = 1 with a > b, c² = a² − b², foci (±c, 0) and eccentricity e = c/a (between 0 and 1). The major axis is 2a and the latus rectum 2b²/a.
Step-by-step method
- Identify a² (larger) and b².
- c = √(a² − b²).
- e = c/a; foci (±c, 0).
Worked example 1 (Basic)
For x²/100 + y²/36 = 1, find the foci and eccentricity.
- c² = 100 − 36 = 64, c = 8
- Foci (±8, 0)
- e = 4/5
- Answer: Foci (±8, 0), e = 4/5
Worked example 2 (Standard)
For x²/169 + y²/25 = 1, find the foci and eccentricity.
- c² = 169 − 25 = 144, c = 12
- Foci (±12, 0)
- e = 12/13
- Answer: Foci (±12, 0), e = 12/13
Worked example 3 (Challenge)
Find the length of the latus rectum of x²/100 + y²/64 = 1.
- 2b²/a = 2 × 64/10
- = 64/5
- Answer: 64/5
Common mistakes and misconceptions
- Using c² = a² + b²
- Taking a as the smaller value
Practice questions
- (Basic) For x²/25 + y²/9 = 1, find the foci and eccentricity.
- (Basic) For x²/25 + y²/16 = 1, find the foci and eccentricity.
- (Standard) For x²/289 + y²/225 = 1, find the foci and eccentricity.
- (Standard) For x²/169 + y²/144 = 1, find the foci and eccentricity.
- (Challenge) Find the length of the latus rectum of x²/289 + y²/225 = 1.
- (Challenge) Find the length of the latus rectum of x²/25 + y²/9 = 1.
Answer key with full working
- Q1: c² = 25 − 9 = 16, c = 4 → Foci (±4, 0) → e = 4/5 → Answer: Foci (±4, 0), e = 4/5
- Q2: c² = 25 − 16 = 9, c = 3 → Foci (±3, 0) → e = 3/5 → Answer: Foci (±3, 0), e = 3/5
- Q3: c² = 289 − 225 = 64, c = 8 → Foci (±8, 0) → e = 8/17 → Answer: Foci (±8, 0), e = 8/17
- Q4: c² = 169 − 144 = 25, c = 5 → Foci (±5, 0) → e = 5/13 → Answer: Foci (±5, 0), e = 5/13
- Q5: 2b²/a = 2 × 225/17 → = 450/17 → Answer: 450/17
- Q6: 2b²/a = 2 × 9/5 → = 18/5 → Answer: 18/5
Syllabus note
This skill is listed under "Conic Sections" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is ellipse: foci and eccentricity in?
"Conic Sections", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.