Class 11 Maths Notes

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Conic Sections Class 11 Notes

Complete, exam-ready notes on conic sections: how slicing a cone produces circles, parabolas, ellipses and hyperbolas, the standard equations for each, foci, eccentricity that classifies them, and latus rectum lengths — written for CBSE boards and JEE revision.

Class:11Subject:MathematicsCovers:CBSE · JEE

7 Key Formulas4 Practice MCQs

DWritten byDeep Narayan

UpdatedSeptember 2026

Key Concept Summary

What is this chapter about in one line?

Conic sections are the four curves — circle, parabola, ellipse, hyperbola — obtained by slicing a double cone at different angles, each defined by eccentricity.

Circles

Definition

Circle

The set of all points at a fixed distance r from a centre (h, k). The standard equation is: $(x - h)^{2} + (y - k)^{2} = r^{2}$ . When centred at the origin (h = k = 0), it simplifies to $x^{2} + y^{2} = r^{2}$ .

General form: x² + y² + 2gx + 2fy + c = 0, with centre (−g, −f) and radius √(g² + f² − c).
A real circle exists only when g² + f² − c > 0.
Eccentricity of a circle is e = 0 — all points are equidistant from the centre.

Parabola

Definition

Parabola

The set of all points equidistant from a fixed point (focus) and a fixed line (directrix). For the standard form $y^{2} = 4 a x$ , the focus is at (a, 0), the directrix is x = −a, and the latus rectum has length 4a.

y² = 4ax opens right; y² = −4ax opens left.
x² = 4ay opens upward; x² = −4ay opens downward.
Eccentricity e = 1 for every parabola.
The latus rectum passes through the focus, perpendicular to the axis.

Ellipse

Definition

Ellipse

The set of all points whose sum of distances from two foci is constant. The standard form (a > b) is: $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ . The major axis has length 2a, the minor axis 2b.

Foci at (±ae, 0) where eccentricity e = √(1 − b²/a²), so 0 < e < 1.
Vertices at (±a, 0); minor-axis vertices at (0, ±b).
Latus rectum length = 2b²/a.
The relation b² = a²(1 − e²) connects a, b and e.

Quick check

If the denominator under x² is larger, the major axis is horizontal. If the denominator under y² is larger, it is vertical — just compare a² and b².

Hyperbola

Definition

Hyperbola

The set of all points whose difference of distances from two foci is constant. The standard form is: $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . The vertices are at (±a, 0) and the foci at (±ae, 0).

Eccentricity e = √(1 + b²/a²), so e > 1 always.
Latus rectum length = 2b²/a.
Asymptotes: y = ±(b/a)x — the curve approaches but never touches these lines.
Conjugate hyperbola: x²/a² − y²/b² = −1 (swap the sign).

Identifying Conics by Eccentricity

e = 0: circle.
e = 1: parabola.
0 < e < 1: ellipse.
e > 1: hyperbola.
This single parameter e classifies all four conics — a key JEE shortcut.

JEE trap

A circle is technically an ellipse with e = 0 and both foci coinciding at the centre. JEE questions occasionally exploit this by asking 'which eccentricity gives a circle?'

Parametric Equations of Conics

Definition

Parametric form

A parametric equation represents each coordinate as a function of a parameter t (or θ), making it easy to plot points or compute tangents. For the standard parabola $y^{2} = 4 a x$ , the parametric form is $x = a t^{2}, y = 2 a t$ .

Ellipse: x = a cos θ, y = b sin θ (θ is the eccentric angle, 0 ≤ θ < 2π).
Hyperbola: x = a sec θ, y = b tan θ (θ is the eccentric angle).
Parametric forms are especially useful in JEE for finding tangent lines and normals.
Substituting the parameter into the standard equation verifies the parametric pair always lies on the curve.

Solved Examples

Example: Find the eccentricity, foci and latus rectum of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ .

Solution: Here a² = 25 (a = 5), b² = 16 (b = 4). Eccentricity e = √(1 − 16/25) = √(9/25) = 3/5. Foci at (±5 × 3/5, 0) = (±3, 0). Latus rectum = 2b²/a = 2(16)/5 = 32/5 units.

Example: A parabola has focus (2, 0) and directrix x = −2. Find its equation.

Solution: The focus is at (a, 0) with a = 2, so the standard form is y² = 4ax. Substituting: y² = 8x.

NCERT Exercise Questions

From the textbook

All NCERT questions for this chapter — NCERT Maths Part I, Chapter 10 (Conic Sections). Questions follow the textbook order so you can match them to the book.

Exercises (NCERT Q1–Q62)

1) Find the equation of the circle with centre (0, 2) and radius 2.
2) Find the equation of the circle with centre (−2, 3) and radius 4.
3) Find the equation of the circle with centre (1/2, 1/4) and radius 1/12.
4) Find the equation of the circle with centre (1, 1) and radius √2.
5) Find the equation of the circle with centre (−a, −b) and radius √(a² − b²).
6) Find the centre and radius of the circle x² + y² + 2x − 4y − 4 = 0.
7) Find the centre and radius of the circle x² + y² − 4x − 8y − 45 = 0.
8) Find the centre and radius of the circle x² + y² − 8x + 10y − 12 = 0.
9) Find the centre and radius of the circle 2x² + 2y² − x = 0.
10) Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.
11) Find the equation of the circle passing through the points (2, 3) and (−1, 1) and whose centre is on the line x − 3y − 11 = 0.
12) Find the equation of the circle with radius 5 whose centre lies on the x-axis and passes through the point (2, 3).
13) Find the equation of the circle passing through (0, 0) and making intercepts a and b on the coordinate axes.
14) Find the equation of a circle with centre (2, 2) and passing through the point (4, 5).
15) Does the point (−2.5, 3.5) lie inside, outside or on the circle x² + y² = 25?
16) Find the coordinates of the focus, the axis, the equation of the directrix and the length of the latus rectum for y² = 12x.
17) Find the coordinates of the focus, the axis, the equation of the directrix and the length of the latus rectum for x² = 6y.
18) Find the coordinates of the focus, the axis, the equation of the directrix and the length of the latus rectum for y² = −8x.
19) Find the coordinates of the focus, the axis, the equation of the directrix and the length of the latus rectum for x² = −16y.
20) Find the coordinates of the focus, the axis, the equation of the directrix and the length of the latus rectum for y² = 10x.
21) Find the coordinates of the focus, the axis, the equation of the directrix and the length of the latus rectum for x² = −9y.
22) Find the equation of the parabola with focus (6, 0) and directrix x = −6.
23) Find the equation of the parabola with focus (0, −3) and directrix y = 3.
24) Find the equation of the parabola with vertex at (0, 0) and focus (3, 0).
25) Find the equation of the parabola with vertex at (0, 0) and focus (−2, 0).
26) Find the equation of the parabola with vertex (0, 0) that passes through (2, 3) and has its axis along the x-axis.
27) Find the equation of the parabola with vertex (0, 0) that passes through (5, 2) and is symmetric about the y-axis.
28) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the ellipse x²/16 + y²/9 = 1.
29) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the ellipse x²/4 + y²/25 = 1.
30) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the ellipse x²/16 + y²/100 = 1.
31) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the ellipse x²/25 + y²/100 = 1.
32) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the ellipse x²/49 + y²/36 = 1.
33) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the ellipse x²/100 + y²/400 = 1.
34) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the ellipse 36x² + 4y² = 144.
35) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the ellipse 16x² + y² = 16.
36) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the ellipse 4x² + 9y² = 36.
37) Find the equation of the ellipse with vertices (±5, 0) and foci (±4, 0).
38) Find the equation of the ellipse with vertices (0, ±13) and foci (0, ±5).
39) Find the equation of the ellipse with vertices (±6, 0) and foci (±4, 0).
40) Find the equation of the ellipse where the ends of the major axis are (±3, 0) and the ends of the minor axis are (0, ±2).
41) Find the equation of the ellipse where the ends of the major axis are (0, ±√5) and the ends of the minor axis are (±1, 0).
42) Find the equation of the ellipse with length of the major axis 26 and foci (±5, 0).
43) Find the equation of the ellipse with length of the major axis 16 and foci (0, ±6).
44) Find the equation of the ellipse with foci (±3, 0), a = 4.
45) Find the equation of the ellipse with b = 3, c = 4, centre at the origin and foci on the x-axis.
46) Find the equation of the ellipse with centre at the origin, major axis on the y-axis and passing through the points (3, 2) and (1, 6).
47) Find the equation of the ellipse with major axis on the x-axis and passing through the points (4, 3) and (6, 2).
48) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola x²/9 − y²/16 = 1.
49) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola y²/9 − x²/27 = 1.
50) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola 9y² − 4x² = 36.
51) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola 16x² − 9y² = 576.
52) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola 5y² − 9x² = 36.
53) Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola 49y² − 16x² = 784.
54) Find the equation of the hyperbola with vertices (±2, 0) and foci (±3, 0).
55) Find the equation of the hyperbola with vertices (0, ±5) and foci (0, ±8).
56) Find the equation of the hyperbola with vertices (0, ±3) and foci (0, ±5).
57) Find the equation of the hyperbola with foci (±5, 0) and the transverse axis of length 8.
58) Find the equation of the hyperbola with foci (0, ±13) and the conjugate axis of length 24.
59) Find the equation of the hyperbola with foci (±3√5, 0) and length of the latus rectum 8.
60) Find the equation of the hyperbola with foci (±4, 0) and length of the latus rectum 12.
61) Find the equation of the hyperbola with vertices (±7, 0) and eccentricity e = 4/3.
62) Find the equation of the hyperbola with foci (0, ±√10) that passes through the point (2, 3).

Answers at a glance

1
Q1 → x² + y² − 4y = 0.
2
Q2 → x² + y² + 4x − 6y − 3 = 0.
3
Q3 → 36x² + 36y² − 36x − 18y + 11 = 0.
4
Q4 → x² + y² − 2x − 2y = 0.
5
Q5 → x² + y² + 2ax + 2by + 2b² = 0.
6
Q6 → Centre (−1, 2), radius 3.
7
Q7 → Centre (2, 4), radius √65.
8
Q8 → Centre (4, −5), radius √53.
9
Q9 → Centre (1/4, 0), radius 1/4.
10
Q10 → x² + y² − 6x − 8y + 15 = 0.
11
Q11 → x² + y² − 7x + 5y − 14 = 0.
12
Q12 → x² + y² − 12x + 11 = 0 or x² + y² + 4x − 21 = 0.
13
Q13 → x² + y² − ax − by = 0.
14
Q14 → x² + y² − 4x − 4y − 5 = 0.
15
Q15 → Inside the circle.
16
Q16 → Focus (3, 0), axis x-axis, directrix x = −3, latus rectum 12.
17
Q17 → Focus (0, 3/2), axis y-axis, directrix y = −3/2, latus rectum 6.
18
Q18 → Focus (−2, 0), axis x-axis, directrix x = 2, latus rectum 8.
19
Q19 → Focus (0, −4), axis y-axis, directrix y = 4, latus rectum 16.
20
Q20 → Focus (5/2, 0), axis x-axis, directrix x = −5/2, latus rectum 10.
21
Q21 → Focus (0, −9/4), axis y-axis, directrix y = 9/4, latus rectum 9.
22
Q22 → y² = 24x.
23
Q23 → x² = −12y.
24
Q24 → y² = 12x.
25
Q25 → y² = −8x.
26
Q26 → y² = (9/2)x.
27
Q27 → x² = (25/2)y.
28
Q28 → Foci (±√7, 0), vertices (±4, 0), e = √7/4, latus rectum 9/2.
29
Q29 → Foci (0, ±√21), vertices (0, ±5), e = √21/5, latus rectum 8/5.
30
Q30 → Foci (0, ±2√21), vertices (0, ±10), e = √21/5, latus rectum 16/5.
31
Q31 → Foci (0, ±5√3), vertices (0, ±10), e = √3/2, latus rectum 5.
32
Q32 → Foci (±√13, 0), vertices (±7, 0), e = √13/7, latus rectum 72/7.
33
Q33 → Foci (0, ±10√3), vertices (0, ±20), e = √3/2, latus rectum 10.
34
Q34 → Foci (0, ±4√2), vertices (0, ±6), e = 2√2/3, latus rectum 4/3.
35
Q35 → Foci (0, ±√15), vertices (0, ±4), e = √15/4, latus rectum 1/2.
36
Q36 → Foci (±√5, 0), vertices (±3, 0), e = √5/3, latus rectum 8/3.
37
Q37 → x²/25 + y²/9 = 1.
38
Q38 → x²/144 + y²/169 = 1.
39
Q39 → x²/36 + y²/20 = 1.
40
Q40 → x²/9 + y²/4 = 1.
41
Q41 → x² + y²/5 = 1.
42
Q42 → x²/169 + y²/144 = 1.
43
Q43 → x²/28 + y²/64 = 1.
44
Q44 → x²/16 + y²/7 = 1.
45
Q45 → x²/25 + y²/9 = 1.
46
Q46 → x²/10 + y²/40 = 1.
47
Q47 → x²/52 + y²/13 = 1.
48
Q48 → Foci (±5, 0), vertices (±3, 0), e = 5/3, latus rectum 32/3.
49
Q49 → Foci (0, ±6), vertices (0, ±3), e = 2, latus rectum 18.
50
Q50 → Foci (0, ±√13), vertices (0, ±2), e = √13/2, latus rectum 9.
51
Q51 → Foci (±10, 0), vertices (±6, 0), e = 5/3, latus rectum 64/3.
52
Q52 → Foci (0, ±2√70/5), vertices (0, ±6/√5), e = √14/3, latus rectum 4√5/3.
53
Q53 → Foci (0, ±√65), vertices (0, ±4), e = √65/4, latus rectum 49/2.
54
Q54 → x²/4 − y²/5 = 1.
55
Q55 → y²/25 − x²/39 = 1.
56
Q56 → y²/9 − x²/16 = 1.
57
Q57 → x²/16 − y²/9 = 1.
58
Q58 → y²/25 − x²/144 = 1.
59
Q59 → x²/25 − y²/20 = 1.
60
Q60 → x²/4 − y²/12 = 1.
61
Q61 → x²/49 − 9y²/343 = 1.
62
Q62 → y²/5 − x²/5 = 1.

CBSE Pattern Questions

How CBSE asks this chapter

CBSE Class 11 Mathematics board paper: Section A (20 MCQs × 1M), Section B (5 × 2M), Section C (6 × 3M) and Section D (4 × 4M). Conic Sections tests 1-mark MCQs on focus, directrix, eccentricity and standard equations, 2/3-mark questions on the parabola, ellipse and hyperbola, and 4-mark questions deriving standard equations from their focus-directrix definitions.

MCQ Questions

The focus of the parabola y² = 4ax is

For an ellipse x²/a² + y²/b² = 1 (a > b), the eccentricity e satisfies

The length of the latus rectum of the parabola y² = 4ax is

The centre of the circle x² + y² = 16 is

1 Mark Questions

1
Write the equation of a circle with centre (0, 0) and radius 5. (Ans: x² + y² = 25)
2
State the eccentricity of a parabola. (Ans: 1)
3
Give the focus of y² = 8x. (Ans: (2, 0))
4
Write the standard equation of an ellipse with foci on the x-axis.

2 Mark Questions

1
Find the equation of the parabola with focus (3, 0) and directrix x = −3. (Ans: y² = 12x)
2
Find the centre and radius of x² + y² − 4x + 6y − 12 = 0. (Ans: centre (2, −3), radius 5)
3
Find the equation of the circle with centre (1, 2) passing through the origin. (Ans: (x − 1)² + (y − 2)² = 5)

3 Mark Questions

1
Find the coordinates of the focus, the equation of the directrix and the length of the latus rectum of the parabola y² = −12x. (Ans: focus (−3, 0), directrix x = 3, latus rectum 12)
2
Find the equation of the ellipse with vertices (±13, 0) and foci (±5, 0). (Ans: x²/169 + y²/144 = 1)
3
Find the equation of the hyperbola with foci (±4, 0) and vertices (±2, 0). (Ans: x²/4 − y²/12 = 1)

4 Mark Questions

1
Define a conic by its focus-directrix property and derive the standard equation y² = 4ax of a parabola whose focus is (a, 0) and directrix x = −a, showing each algebra step.
2
Derive the equation of an ellipse with foci (−c, 0), (c, 0) and the sum of focal distances 2a, arriving at x²/a² + y²/(a² − c²) = 1, and state its eccentricity. (Ans: e = c/a)

PDF: Assignments & Tests

Printable study material

Free downloadable PDFs for Conic Sections — two topic-wise assignments covering every concept of the chapter, and two full CBSE-pattern tests (each 90 minutes, 40 marks) mirroring the board paper. No signup required.

Free PDF Downloads & Assignments

No Signup Required

Practice path: complete Assignment 1 first (topic-by-topic), then Assignment 2 (mixed difficulty with full solutions), then attempt the timed tests.

Conic Sections Assignment 1 — Topic-wise Questions

Covers all six topics: circles; parabola; ellipse; hyperbola; identifying conics by eccentricity; and parametric equations of conics.

conic-sections-class-11-notes-assignment-1.pdf

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Conic Sections Assignment 2 — Challenge + Integrated Problems

Higher-difficulty set with multi-step problems (circle through points, focus/directrix/eccentricity, latus rectum, conic identification, parametric points) and full worked answers. Includes an answer key.

conic-sections-class-11-notes-assignment-2.pdf

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Conic Sections Test 1 — CBSE Pattern (90 min, 40 marks)

Full board-pattern test: Section A (MCQ/VSA 1M × 10), Section B (2M × 4), Section C (3M × 4), Section D (5M × 2) = 40 marks. Time: 90 minutes. Includes answer key.

conic-sections-class-11-notes-test-1-cbse.pdf

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Conic Sections Test 2 — CBSE Pattern (90 min, 40 marks)

Second full paper with different questions, same 40-mark / 90-minute blueprint. Includes answer key.

conic-sections-class-11-notes-test-2-cbse.pdf

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Quick Revision

Key formulas at a glance

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

Circle (centred at origin)

x^{2} + y^{2} = r^{2}

Circle (general centre)

(x - h)^{2} + (y - k)^{2} = r^{2}

Parabola

y^{2} = 4 a x

Ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

Ellipse eccentricity

e = 1 - \frac{b ^{2}}{a ^{2}}

Hyperbola

\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1

Hyperbola eccentricity

e = 1 + \frac{b ^{2}}{a ^{2}}

Exam Strategy

How this chapter is asked

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

Circle has e = 0, parabola e = 1, ellipse 0 < e < 1, hyperbola e > 1.
Parabola latus rectum = 4a; always the perpendicular chord through the focus.
Ellipse: b² = a²(1 − e²); this relation connects a, b and e.
Hyperbola eccentricity is always > 1; never forget the + sign under the root.
Identify the conic first from the equation before applying formulas — saves time.
For latus rectum of ellipse and hyperbola, both use 2b²/a — but eccentricity formulas differ.
General second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0: check discriminant B² − 4AC to classify.
JEE loves shifted conics — complete the square to bring any conic to standard form.

FAQ

Frequently asked questions

What is eccentricity?

Eccentricity e is a number that classifies a conic: e = 0 is a circle, e = 1 is a parabola, 0 < e < 1 is an ellipse, and e > 1 is a hyperbola.

What is the latus rectum?

The latus rectum is a chord through the focus, perpendicular to the axis of symmetry. For a parabola it is 4a; for an ellipse and hyperbola it is 2b²/a.

How do you find the focus of a parabola from its equation?

For y² = 4ax, the focus is at (a, 0) and directrix is x = −a. Identify a by comparing your equation with this standard form.

What is the difference between an ellipse and a hyperbola?

An ellipse has 0 < e < 1 and a + b sum of focal distances; a hyperbola has e > 1 and a constant difference of focal distances. Their equations differ by a sign.

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