ClassApna

Class 11 Maths NCERT Solutions

~15 min read

Conic Sections Class 11 Maths NCERT Solutions

The complete NCERT exercise solutions for Chapter 10, Conic Sections — 62 questions from Ex 10.1 to Ex 10.4, each worked through step by step in the CBSE marking pattern. Circles, parabolas, ellipses and hyperbolas, their standard forms, foci, directrices, eccentricity and latus rectum.

Class:11Subject:MathsChapter:10
4 Key Formulas
DWritten byDeep Narayan
Updated
Key Concept Summary

How many questions are in NCERT Class 11 Maths Chapter 10?

Chapter 10 carries 4 exercise questions, numbered Ex 10.1 to Ex 10.4. All of them are solved step by step on this page, along with the chapter's key formulas and exam pointers.

01

Chapter Overview

The four curves formed by slicing a double cone — circle, parabola, ellipse and hyperbola — are the conic sections. A circle keeps points equidistant from a centre. A parabola keeps a point equidistant from a fixed point (focus) and a fixed line (directrix). An ellipse keeps the sum of distances to two foci constant, and a hyperbola keeps the difference constant. This chapter writes each in standard form and reads off its geometric features: foci, vertices, eccentricity and latus rectum.

Board pattern

For a parabola, the latus rectum has length 4a; for y² = 4ax the ends are (±2a, a)-style. For an ellipse, c² = a² − b², eccentricity e = c/a (< 1), latus rectum 2b²/a. For a hyperbola, c² = a² + b², e = c/a (> 1), latus rectum 2b²/a. When the major axis is on the y-axis, swap a and b in the denominators — and remember which axis is long.
02

Exercise 10.1 — The Circle

15Exercise questions

Step-by-step solution

  1. 1(x − 0)² + (y − 2)² = 2².
  2. 2x² + y² − 4y = 0.

Final answer

x² + y² − 4y = 0.

Step-by-step solution

  1. 1(x + 2)² + (y − 3)² = 16.
  2. 2x² + y² + 4x − 6y − 3 = 0.

Final answer

x² + y² + 4x − 6y − 3 = 0.

Step-by-step solution

  1. 1(x − 1/2)² + (y − 1/4)² = 1/144.
  2. 2Multiply by 144: 144x² − 144x + 36 + 144y² − 72y + 9 = 1.
  3. 336x² + 36y² − 36x − 18y + 11 = 0.

Final answer

36x² + 36y² − 36x − 18y + 11 = 0.

Step-by-step solution

  1. 1(x − 1)² + (y − 1)² = 2.
  2. 2x² + y² − 2x − 2y = 0.

Final answer

x² + y² − 2x − 2y = 0.

Step-by-step solution

  1. 1(x + a)² + (y + b)² = a² − b².
  2. 2x² + 2ax + a² + y² + 2by + b² = a² − b².
  3. 3x² + y² + 2ax + 2by + 2b² = 0.

Final answer

x² + y² + 2ax + 2by + 2b² = 0.

Step-by-step solution

  1. 1Complete squares: (x + 1)² + (y − 2)² = 4 + 1 + 4 = 9.
  2. 2Centre (−1, 2), radius 3.

Final answer

Centre (−1, 2), radius 3.

Step-by-step solution

  1. 1(x − 2)² + (y − 4)² = 45 + 4 + 16 = 65.
  2. 2Centre (2, 4), radius √65.

Final answer

Centre (2, 4), radius √65.

Step-by-step solution

  1. 1(x − 4)² + (y + 5)² = 12 + 16 + 25 = 53.
  2. 2Centre (4, −5), radius √53.

Final answer

Centre (4, −5), radius √53.

Step-by-step solution

  1. 1Divide by 2: x² + y² − x/2 = 0.
  2. 2(x − 1/4)² + y² = 1/16.
  3. 3Centre (1/4, 0), radius 1/4.

Final answer

Centre (1/4, 0), radius 1/4.

Step-by-step solution

  1. 1Let centre be (h, k): (h − 4)² + (k − 1)² = (h − 6)² + (k − 5)² → h + 2k = 11.
  2. 2With 4h + k = 16: solve → h = 3, k = 4.
  3. 3r² = (3 − 4)² + (4 − 1)² = 10.
  4. 4(x − 3)² + (y − 4)² = 10 → x² + y² − 6x − 8y + 15 = 0.

Final answer

x² + y² − 6x − 8y + 15 = 0.

Step-by-step solution

  1. 1Equal radii: (h − 2)² + (k − 3)² = (h + 1)² + (k − 1)² → 6h + 4k = 11.
  2. 2With h − 3k = 11: solve → h = 7/2, k = −5/2.
  3. 3r² = (7/2 − 2)² + (−5/2 − 3)² = 65/2.
  4. 4(x − 7/2)² + (y + 5/2)² = 65/2 → x² + y² − 7x + 5y − 14 = 0.

Final answer

x² + y² − 7x + 5y − 14 = 0.

Step-by-step solution

  1. 1Centre (h, 0): (2 − h)² + 9 = 25 → (h − 2)² = 16 → h = 6 or −2.
  2. 2h = 6: (x − 6)² + y² = 25 → x² + y² − 12x + 11 = 0.
  3. 3h = −2: (x + 2)² + y² = 25 → x² + y² + 4x − 21 = 0.

Final answer

x² + y² − 12x + 11 = 0 or x² + y² + 4x − 21 = 0.

Step-by-step solution

  1. 1General form x² + y² + 2gx + 2fy + c = 0; through (0,0) gives c = 0.
  2. 2x-intercepts come from x² + 2gx = 0 → 0 and −2g; so −2g = a.
  3. 3Similarly y-intercepts 0 and −2f = b.
  4. 4x² + y² − ax − by = 0.

Final answer

x² + y² − ax − by = 0.

Step-by-step solution

  1. 1r² = (4 − 2)² + (5 − 2)² = 13.
  2. 2(x − 2)² + (y − 2)² = 13 → x² + y² − 4x − 4y − 5 = 0.

Final answer

x² + y² − 4x − 4y − 5 = 0.

Step-by-step solution

  1. 1(−2.5)² + (3.5)² = 6.25 + 12.25 = 18.5.
  2. 218.5 < 25 → inside the circle.

Final answer

Inside the circle.

03

Exercise 10.2 — The Parabola

12Exercise questions

Step-by-step solution

  1. 1Compare with y² = 4ax: 4a = 12 → a = 3 (opens right).
  2. 2Focus (3, 0), axis y = 0, directrix x = −3.
  3. 3Latus rectum = 4a = 12.

Final answer

Focus (3, 0), axis x-axis, directrix x = −3, latus rectum 12.

Step-by-step solution

  1. 1Compare with x² = 4ay: 4a = 6 → a = 3/2 (opens up).
  2. 2Focus (0, 3/2), axis x = 0, directrix y = −3/2.
  3. 3Latus rectum = 6.

Final answer

Focus (0, 3/2), axis y-axis, directrix y = −3/2, latus rectum 6.

Step-by-step solution

  1. 1y² = −4ax with 4a = 8 → a = 2 (opens left).
  2. 2Focus (−2, 0), axis y = 0, directrix x = 2.
  3. 3Latus rectum = 8.

Final answer

Focus (−2, 0), axis x-axis, directrix x = 2, latus rectum 8.

Step-by-step solution

  1. 1x² = −4ay with 4a = 16 → a = 4 (opens down).
  2. 2Focus (0, −4), axis x = 0, directrix y = 4.
  3. 3Latus rectum = 16.

Final answer

Focus (0, −4), axis y-axis, directrix y = 4, latus rectum 16.

Step-by-step solution

  1. 1y² = 4ax with 4a = 10 → a = 5/2 (opens right).
  2. 2Focus (5/2, 0), axis y = 0, directrix x = −5/2.
  3. 3Latus rectum = 10.

Final answer

Focus (5/2, 0), axis x-axis, directrix x = −5/2, latus rectum 10.

Step-by-step solution

  1. 1x² = −4ay with 4a = 9 → a = 9/4 (opens down).
  2. 2Focus (0, −9/4), axis x = 0, directrix y = 9/4.
  3. 3Latus rectum = 9.

Final answer

Focus (0, −9/4), axis y-axis, directrix y = 9/4, latus rectum 9.

Step-by-step solution

  1. 1Focus on positive x-axis with |directrix| = 6 → a = 6.
  2. 2y² = 4ax = 24x.

Final answer

y² = 24x.

Step-by-step solution

  1. 1Focus (0, −3) ⇒ axis along the y-axis opening down, a = 3.
  2. 2x² = −4ay = −12y.

Final answer

x² = −12y.

Step-by-step solution

  1. 1Focus (3, 0) ⇒ axis along x-axis, a = 3.
  2. 2y² = 4ax = 12x.

Final answer

y² = 12x.

Step-by-step solution

  1. 1Focus (−2, 0) ⇒ axis along x-axis opening left, a = 2.
  2. 2y² = −4ax = −8x.

Final answer

y² = −8x.

Step-by-step solution

  1. 1Axis along x-axis: y² = 4ax.
  2. 2(3)² = 4a(2) → a = 9/8.
  3. 3y² = 4(9/8)x → y² = (9/2)x.

Final answer

y² = (9/2)x.

Step-by-step solution

  1. 1Symmetric about y-axis: x² = 4ay.
  2. 2(5)² = 4a(2) → a = 25/8.
  3. 3x² = (25/2)y.

Final answer

x² = (25/2)y.

04

Exercise 10.3 — The Ellipse

20Exercise questions

Step-by-step solution

  1. 1a² = 16, b² = 9 → a = 4, b = 3, c = √(16 − 9) = √7.
  2. 2Major axis along x-axis: vertices (±4, 0), foci (±√7, 0).
  3. 3e = c/a = √7/4; latus rectum = 2b²/a = 18/4 = 9/2.

Final answer

Foci (±√7, 0), vertices (±4, 0), e = √7/4, latus rectum 9/2.

Step-by-step solution

  1. 1a² = 25, b² = 4 → a = 5, b = 2, c = √21 (major axis on y-axis).
  2. 2Vertices (0, ±5), foci (0, ±√21).
  3. 3e = √21/5; latus rectum = 2b²/a = 8/5.

Final answer

Foci (0, ±√21), vertices (0, ±5), e = √21/5, latus rectum 8/5.

Step-by-step solution

  1. 1a² = 100, b² = 16 → a = 10, b = 4, c = √84 = 2√21.
  2. 2Vertices (0, ±10), foci (0, ±2√21).
  3. 3e = 2√21/10 = √21/5; latus rectum = 2·16/10 = 16/5.

Final answer

Foci (0, ±2√21), vertices (0, ±10), e = √21/5, latus rectum 16/5.

Step-by-step solution

  1. 1a² = 100, b² = 25 → a = 10, b = 5, c = 5√3.
  2. 2Vertices (0, ±10), foci (0, ±5√3).
  3. 3e = √3/2; latus rectum = 2·25/10 = 5.

Final answer

Foci (0, ±5√3), vertices (0, ±10), e = √3/2, latus rectum 5.

Step-by-step solution

  1. 1a² = 49, b² = 36 → a = 7, b = 6, c = √13.
  2. 2Vertices (±7, 0), foci (±√13, 0).
  3. 3e = √13/7; latus rectum = 2·36/7 = 72/7.

Final answer

Foci (±√13, 0), vertices (±7, 0), e = √13/7, latus rectum 72/7.

Step-by-step solution

  1. 1a² = 400, b² = 100 → a = 20, b = 10, c = 10√3.
  2. 2Vertices (0, ±20), foci (0, ±10√3).
  3. 3e = √3/2; latus rectum = 2·100/20 = 10.

Final answer

Foci (0, ±10√3), vertices (0, ±20), e = √3/2, latus rectum 10.

Step-by-step solution

  1. 1Divide by 144: x²/4 + y²/36 = 1 → a = 6, b = 2, c = 4√2.
  2. 2Vertices (0, ±6), foci (0, ±4√2).
  3. 3e = 2√2/3; latus rectum = 2·4/6 = 4/3.

Final answer

Foci (0, ±4√2), vertices (0, ±6), e = 2√2/3, latus rectum 4/3.

Step-by-step solution

  1. 1x²/1 + y²/16 = 1 → a = 4, b = 1, c = √15.
  2. 2Vertices (0, ±4), foci (0, ±√15).
  3. 3e = √15/4; latus rectum = 2·1/4 = 1/2.

Final answer

Foci (0, ±√15), vertices (0, ±4), e = √15/4, latus rectum 1/2.

Step-by-step solution

  1. 1x²/9 + y²/4 = 1 → a = 3, b = 2, c = √5.
  2. 2Vertices (±3, 0), foci (±√5, 0).
  3. 3e = √5/3; latus rectum = 2·4/3 = 8/3.

Final answer

Foci (±√5, 0), vertices (±3, 0), e = √5/3, latus rectum 8/3.

Step-by-step solution

  1. 1a = 5, c = 4 → b² = 25 − 16 = 9.
  2. 2x²/25 + y²/9 = 1.

Final answer

x²/25 + y²/9 = 1.

Step-by-step solution

  1. 1a = 13, c = 5 → b² = 169 − 25 = 144.
  2. 2Major axis on y-axis: x²/144 + y²/169 = 1.

Final answer

x²/144 + y²/169 = 1.

Step-by-step solution

  1. 1a = 6, c = 4 → b² = 36 − 16 = 20.
  2. 2x²/36 + y²/20 = 1.

Final answer

x²/36 + y²/20 = 1.

Step-by-step solution

  1. 1Major on x-axis with a = 3, b = 2.
  2. 2x²/9 + y²/4 = 1.

Final answer

x²/9 + y²/4 = 1.

Step-by-step solution

  1. 1Major on y-axis with a = √5, b = 1.
  2. 2x²/1 + y²/5 = 1.

Final answer

x² + y²/5 = 1.

Step-by-step solution

  1. 12a = 26 → a = 13; c = 5 → b² = 169 − 25 = 144.
  2. 2x²/169 + y²/144 = 1.

Final answer

x²/169 + y²/144 = 1.

Step-by-step solution

  1. 12a = 16 → a = 8; c = 6 → b² = 64 − 36 = 28.
  2. 2Major axis on y-axis: x²/28 + y²/64 = 1.

Final answer

x²/28 + y²/64 = 1.

Step-by-step solution

  1. 1c = 3, a = 4 → b² = 16 − 9 = 7.
  2. 2x²/16 + y²/7 = 1.

Final answer

x²/16 + y²/7 = 1.

Step-by-step solution

  1. 1a² = b² + c² = 9 + 16 = 25 → a = 5.
  2. 2x²/25 + y²/9 = 1.

Final answer

x²/25 + y²/9 = 1.

Step-by-step solution

  1. 1Form x²/b² + y²/a² = 1: 9/b² + 4/a² = 1 and 1/b² + 36/a² = 1.
  2. 2Eliminate: 36(1 − 9/b²) = 4(1 − 1/b²) → b² = 10.
  3. 3Then 9/10 + 4/a² = 1 → a² = 40.
  4. 4x²/10 + y²/40 = 1.

Final answer

x²/10 + y²/40 = 1.

Step-by-step solution

  1. 1Form x²/a² + y²/b² = 1 with x = 1/a², y = 1/b²: 16x + 9y = 1 and 36x + 4y = 1.
  2. 2Solve: x = 1/52, y = 1/13 → a² = 52, b² = 13.
  3. 3x²/52 + y²/13 = 1.

Final answer

x²/52 + y²/13 = 1.

05

Exercise 10.4 — The Hyperbola

15Exercise questions

Step-by-step solution

  1. 1a = 3, b = 4, c = √(9 + 16) = 5.
  2. 2Transverse axis along x-axis: vertices (±3, 0), foci (±5, 0).
  3. 3e = 5/3; latus rectum = 2b²/a = 32/3.

Final answer

Foci (±5, 0), vertices (±3, 0), e = 5/3, latus rectum 32/3.

Step-by-step solution

  1. 1a = 3, b = 3√3, c = √(9 + 27) = 6.
  2. 2Transverse axis along y-axis: vertices (0, ±3), foci (0, ±6).
  3. 3e = 2; latus rectum = 2·27/3 = 18.

Final answer

Foci (0, ±6), vertices (0, ±3), e = 2, latus rectum 18.

Step-by-step solution

  1. 1y²/4 − x²/9 = 1 → a = 2, b = 3, c = √13.
  2. 2Vertices (0, ±2), foci (0, ±√13).
  3. 3e = √13/2; latus rectum = 2·9/2 = 9.

Final answer

Foci (0, ±√13), vertices (0, ±2), e = √13/2, latus rectum 9.

Step-by-step solution

  1. 1x²/36 − y²/64 = 1 → a = 6, b = 8, c = 10.
  2. 2Vertices (±6, 0), foci (±10, 0).
  3. 3e = 5/3; latus rectum = 2·64/6 = 64/3.

Final answer

Foci (±10, 0), vertices (±6, 0), e = 5/3, latus rectum 64/3.

Step-by-step solution

  1. 1y²/(36/5) − x²/4 = 1 → a = 6/√5, b = 2, c = √(56/5) = 2√70/5.
  2. 2Vertices (0, ±6/√5), foci (0, ±2√70/5).
  3. 3e = c/a = √14/3; latus rectum = 2b²/a = 4√5/3.

Final answer

Foci (0, ±2√70/5), vertices (0, ±6/√5), e = √14/3, latus rectum 4√5/3.

Step-by-step solution

  1. 1y²/16 − x²/49 = 1 → a = 4, b = 7, c = √65.
  2. 2Vertices (0, ±4), foci (0, ±√65).
  3. 3e = √65/4; latus rectum = 2·49/4 = 49/2.

Final answer

Foci (0, ±√65), vertices (0, ±4), e = √65/4, latus rectum 49/2.

Step-by-step solution

  1. 1a = 2, c = 3 → b² = 9 − 4 = 5.
  2. 2x²/4 − y²/5 = 1.

Final answer

x²/4 − y²/5 = 1.

Step-by-step solution

  1. 1a = 5 (on y-axis), c = 8 → b² = 64 − 25 = 39.
  2. 2y²/25 − x²/39 = 1.

Final answer

y²/25 − x²/39 = 1.

Step-by-step solution

  1. 1a = 3, c = 5 → b² = 25 − 9 = 16.
  2. 2y²/9 − x²/16 = 1.

Final answer

y²/9 − x²/16 = 1.

Step-by-step solution

  1. 12a = 8 → a = 4; c = 5 → b² = 25 − 16 = 9.
  2. 2x²/16 − y²/9 = 1.

Final answer

x²/16 − y²/9 = 1.

Step-by-step solution

  1. 12b = 24 → b = 12; c = 13 → a² = 169 − 144 = 25.
  2. 2y²/25 − x²/144 = 1.

Final answer

y²/25 − x²/144 = 1.

Step-by-step solution

  1. 1c² = 45 and latus rectum 2b²/a = 8 → b² = 4a.
  2. 2a² + 4a = 45 → a² + 4a − 45 = 0 → a = 5 (b² = 20).
  3. 3x²/25 − y²/20 = 1.

Final answer

x²/25 − y²/20 = 1.

Step-by-step solution

  1. 1c = 4 and 2b²/a = 12 → b² = 6a.
  2. 2a² + 6a = 16 → a² + 6a − 16 = 0 → a = 2 (b² = 12).
  3. 3x²/4 − y²/12 = 1.

Final answer

x²/4 − y²/12 = 1.

Step-by-step solution

  1. 1a = 7 → c = ae = 28/3 → b² = c² − a² = 343/9.
  2. 2x²/49 − y²/(343/9) = 1.

Final answer

x²/49 − 9y²/343 = 1.

Step-by-step solution

  1. 1c² = 10 = a² + b²; form y²/a² − x²/b² = 1.
  2. 2At (2, 3): 9/a² − 4/b² = 1 → with b² = 10 − a² get a⁴ − 23a² + 90 = 0.
  3. 3a² = 5 (as a² < 10), b² = 5.
  4. 4y²/5 − x²/5 = 1.

Final answer

y²/5 − x²/5 = 1.

Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Parabola

Ellipse

Hyperbola

Latus rectum

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

  • The sign difference in c² separates the conics: a² − b² for an ellipse, a² + b² for a hyperbola, and c = a with e = 1 for a parabola.
  • Identify the conic from its eccentricity first — e = 1 parabola, e < 1 ellipse, e > 1 hyperbola — then use the matching standard form.

FAQ

Frequently asked questions

How many questions are in NCERT Class 11 Maths Chapter 10 (Conic Sections)?

There are 4 exercise questions in this chapter, numbered Ex 10.1 to Ex 10.4. Every one is solved step by step on this page in the official NCERT numbering.

Which formulas come up in Conic Sections Class 11 Maths?

The formulas this chapter's questions actually turn on are: Parabola, Ellipse, Hyperbola, Latus rectum. They are listed with their expressions in the key formulas section below, and the solved questions show where each one is used.

Is Conic Sections important for JEE Main?

Very important — conic sections are a major board unit and a regular JEE Main topic, and the focus-directrix definitions are frequently examined directly.

Next Chapters

Continue learning

Previous: Straight LinesNext: Introduction to Three Dimensional GeometryConic Sections revision notesAll Class 11 Maths NCERT Solutions

Same solutions, live doubt-clearing help

Reading a solution is step one — getting a doubt resolved in real time is what clears it. ClassApna runs small-batch CBSE, JEE & NEET coaching with daily doubt sessions and mock tests.

Book a Free Demo ClassExplore Plans & Batches

Small batches · 1-on-1 personal mentorship · Live online & offline centre