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CBSE Class 11 Mathematics★ 6-8 marks · CBSE Class 11 Mathematics Chapter 10

Conic Sections (Circles, Ellipses, Parabolas, Hyperbolas)

Slice a cone at different angles — you get a CIRCLE, an ELLIPSE, a PARABOLA, or a HYPERBOLA. These four curves describe planetary orbits, satellite dishes, and the paths of projectiles. CBSE Class 11 Mathematics Chapter 10.

⏱ 28 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Write and identify the standard equation of each conic (circle, parabola, ellipse, hyperbola)
2Find focus, directrix, vertices, and latus rectum for any conic in standard form
3State and apply eccentricity values to classify conics
4Solve problems involving conics given partial information (e.g., find equation given focus and directrix)

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Why this chapter matters

Conic sections appear in JEE, CBSE boards, and real-world applications (satellite orbits are ellipses, parabolic reflectors, hyperbolic navigation). Mastering all four conics — their equations, foci, directrix, eccentricity, and latus rectum — is essential for Class 12 and competitive exams.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Straight Lines

Coordinate geometry foundation

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Relations and Functions

Function notation used throughout

Conic Sections

"The same cone, sliced four ways, produces the four curves that describe the cosmos."

1. Chapter Overview

CONIC SECTIONS are the curves obtained by slicing a double-napped right circular cone with a plane at different angles. The four curves are: CIRCLE, ELLIPSE, PARABOLA, and HYPERBOLA. This chapter covers the standard equations, key parameters (centre, radius, focus, directrix, eccentricity, latus rectum), and applications of each.

2. How Conics Are Formed

Plane cuts the cone...	Resulting Curve
Parallel to the BASE	CIRCLE
At an angle (not parallel, not through vertex)	ELLIPSE
Parallel to a GENERATOR (side)	PARABOLA
Cuts BOTH nappes	HYPERBOLA

3. Circle

Definition

Set of all points in a plane that are at a FIXED DISTANCE (radius) from a FIXED POINT (centre)

Standard Equation

Centre at (h, k), radius r: (x — h)² + (y — k)² = r²
Centre at origin (0, 0): x² + y² = r²

General Equation

x² + y² + 2gx + 2fy + c = 0
Centre = (-g, -f). Radius = √(g² + f² — c)
Represents a real circle only if g² + f² — c > 0

4. Parabola

Definition

Set of all points in a plane EQUIDISTANT from a fixed point (FOCUS) and a fixed line (DIRECTRIX)

Key Terms

Focus (F) : The fixed point
Directrix: The fixed line
Axis: The line through the focus, perpendicular to the directrix
Vertex: The midpoint between focus and directrix (where the parabola TURNS)
Latus Rectum: A chord through the focus, perpendicular to the axis. Length = 4a.
Eccentricity (e) : For a parabola, e = 1 (exactly). The defining ratio.

Standard Equations

Form	Equation	Focus	Directrix	Axis	Opens
Right	y² = 4ax	(a, 0)	x = -a	x-axis	RIGHT
Left	y² = -4ax	(-a, 0)	x = a	x-axis	LEFT
Up	x² = 4ay	(0, a)	y = -a	y-axis	UPWARD
Down	x² = -4ay	(0, -a)	y = a	y-axis	DOWNWARD

5. Ellipse

Definition

Set of all points in a plane whose SUM of distances from TWO FIXED POINTS (foci) is CONSTANT
For any point P on the ellipse: PF₁ + PF₂ = 2a (constant = length of MAJOR AXIS)

Key Terms

Centre: Midpoint of the two foci
Major Axis: The LONGEST chord through centre. Length = 2a.
Minor Axis: Perpendicular to major. Length = 2b.
Eccentricity e = c/a, where c = distance from centre to focus. 0 < e < 1.
c² = a² — b² (for a > b)
Latus Rectum: Length = 2b²/a

Standard Equation

Centre at origin. Major axis along x-axis (a > b): x²/a² + y²/b² = 1
Major axis along y-axis (b > a): x²/a² + y²/b² = 1 (here b > a, major axis vertical)

Special Case

If a = b → e = 0 → CIRCLE (a circle is a special case of an ellipse with zero eccentricity)

6. Hyperbola

Definition

Set of all points in a plane whose DIFFERENCE of distances from TWO FIXED POINTS (foci) is CONSTANT
For any point P: |PF₁ — PF₂| = 2a

Key Terms

Transverse Axis: The line through the foci. Length = 2a.
Conjugate Axis: Perpendicular to transverse. Length = 2b.
Eccentricity e = c/a, where c² = a² + b². e > 1.
Asymptotes: Lines the hyperbola approaches but never touches
Latus Rectum: Length = 2b²/a

Standard Equation

Centre at origin. Transverse axis along x-axis: x²/a² — y²/b² = 1
Transverse axis along y-axis: y²/a² — x²/b² = 1

7. Comparison of the Four Conics

Conic	Eccentricity (e)	Defining Property
Circle	e = 0	Distance from centre = constant
Ellipse	0 < e < 1	Sum of distances from 2 foci = constant
Parabola	e = 1	Distance from focus = distance from directrix
Hyperbola	e > 1	Difference of distances from 2 foci = constant

8. Exam Focus

Circle — standard and general equations, finding centre and radius
Parabola — 4 standard forms (y²=4ax, etc.), focus, directrix, latus rectum
Ellipse — standard equation, eccentricity, latus rectum (2b²/a), c² = a² — b²
Hyperbola — standard equation, c² = a² + b², asymptotes
Eccentricity — for each conic, what e tells you

9. Conclusion

Four curves, one cone, universal applications:

CIRCLE: e = 0. Wheels, orbits, arches.
PARABOLA: e = 1. Projectile paths. Satellite dishes. Reflectors.
ELLIPSE: e < 1. Planetary orbits (Kepler's First Law).
HYPERBOLA: e > 1. Some comets. LORAN navigation.

'There is no branch of mathematics, however abstract, which may not someday be applied to phenomena of the real world.' — Lobachevsky. The conics prove him right: abstract Greek geometry → Kepler's laws → space travel.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Circle (standard)

(x − h)² + (y − k)² = r²

Centre (h, k), radius r. Origin form: x² + y² = r²

Parabola (rightward)

y² = 4ax

Focus (a, 0), Directrix x = −a, Latus rectum = 4a, Vertex (0,0)

Parabola (upward)

x² = 4ay

Focus (0, a), Directrix y = −a

Ellipse (major along x)

x²/a² + y²/b² = 1, where a > b > 0

c² = a² − b², Foci (±c, 0), e = c/a (0 < e < 1), Latus rectum = 2b²/a

Ellipse (major along y)

x²/b² + y²/a² = 1, where a > b > 0

c² = a² − b², Foci (0, ±c)

Hyperbola (x-axis)

x²/a² − y²/b² = 1

c² = a² + b², Foci (±c, 0), e = c/a (e > 1), Latus rectum = 2b²/a, Asymptotes: y = ±(b/a)x

Hyperbola (y-axis)

y²/a² − x²/b² = 1

c² = a² + b², Foci (0, ±c)

Eccentricity summary

Circle: e=0 | Ellipse: 0<e<1 | Parabola: e=1 | Hyperbola: e>1

e = c/a for all conics

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Confusing c²=a²−b² (ellipse) with c²=a²+b² (hyperbola)

✓ Ellipse: c² = a² MINUS b² (since c < a). Hyperbola: c² = a² PLUS b² (since c > a). The sign flips because the relationship is different.

WATCH OUT

✗ Using a > b for ellipse without checking which axis is major

✓ Always check which denominator is larger. If a² > b², major axis is along x-axis and a is under x². If a² is under y², major axis is along y-axis.

WATCH OUT

✗ Mixing up the parabola y²=4ax (opens right) vs x²=4ay (opens up)

✓ If y is squared → parabola opens left/right. If x is squared → opens up/down. Positive a = opens toward positive axis.

WATCH OUT

✗ Latus rectum = 2a instead of 4a for parabola

✓ For parabola y²=4ax, the full latus rectum length is 4a (not 2a). For ellipse it is 2b²/a. These are different — memorise separately.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 10.1

Exercise 10.1

Circles — find equation given centre/radius, find centre/radius from equation

Questions

Ex 10.2

Exercise 10.2

Parabolas — find focus, directrix, length of latus rectum; find equation from given conditions

Questions

Ex 10.3

Exercise 10.3

Ellipses — find vertices, foci, eccentricity, latus rectum; write equation from given properties

Questions

Ex 10.4

Exercise 10.4

Hyperbolas — find vertices, foci, eccentricity, latus rectum, asymptotes

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Circle

Find the equation of a circle with centre (2, −3) and radius 4.

▶ Show solution

(x−2)² + (y+3)² = 16

Q2MEDIUM· Parabola

Find the focus, directrix, and length of latus rectum of y² = 12x.

▶ Show solution

Comparing y²=4ax: 4a=12 → a=3. Focus = (3, 0). Directrix: x = −3. Length of latus rectum = 4a = 12.

Q3MEDIUM· Ellipse

Find the equation of the ellipse with foci (±3, 0) and a = 5.

▶ Show solution

c=3, a=5 → b²=a²−c²=25−9=16. Equation: x²/25 + y²/16 = 1.

Q4HARD· Hyperbola

For the hyperbola x²/9 − y²/16 = 1, find eccentricity, foci, and equations of asymptotes.

▶ Show solution

a²=9, b²=16 → c²=a²+b²=25 → c=5. e=c/a=5/3. Foci: (±5, 0). Asymptotes: y=±(b/a)x = ±(4/3)x.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Circle: (x−h)²+(y−k)²=r², e=0
•Parabola y²=4ax: opens right, focus (a,0), directrix x=−a, LR=4a, e=1
•Ellipse x²/a²+y²/b²=1 (a>b): c²=a²−b², foci (±c,0), e=c/a<1, LR=2b²/a
•Hyperbola x²/a²−y²/b²=1: c²=a²+b², foci (±c,0), e=c/a>1, LR=2b²/a, asymptotes y=±(b/a)x
•Eccentricity: circle 0, ellipse (0,1), parabola 1, hyperbola (>1)
•Latus rectum: parabola=4a, ellipse=2b²/a, hyperbola=2b²/a

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-8 marks

Question type	Marks each	Typical count	What it tests
Short/MCQ	1-2	2	Identify conic, find eccentricity, classify
Long	4-6	1-2	Find all parameters of a given conic; write equation from conditions

Prep strategy

Memorise all 8 standard forms (each conic has 2 orientations)
Learn the parameter hierarchy: circle → parabola → ellipse → hyperbola
Practice identifying which form applies before computing
JEE aspirants: conics appear in ~3-4 questions per paper

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Planetary orbits

All planets orbit the Sun in ELLIPSES (Kepler's First Law). The Sun is at one focus. Earth's orbit has eccentricity ≈ 0.017 — nearly circular.

Parabolic reflectors

Satellite dishes, car headlights, and telescope mirrors are all parabolic. A parabola focuses all parallel rays to its focus — making reflectors extremely efficient.

Hyperbolic navigation (LORAN)

Radio navigation systems use the property that the difference in distances from two transmitters to a receiver traces a hyperbola. The receiver's location is where two hyperbolas intersect.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Write the standard form first, then identify a, b, c by comparison — never guess
For 'find the equation' problems, identify what's given (focus, directrix, vertices, eccentricity) and use the definition
For JEE: conic section problems often combine two conics or give the general second-degree equation — practice identifying type from Ax²+Bxy+Cy²+...
Always draw a rough sketch — it prevents orientation errors

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Derive the general second-degree equation Ax²+Bxy+Cy²+Dx+Ey+F=0 and the discriminant B²−4AC for classifying conics
Parametric forms: circle (r cosθ, r sinθ), ellipse (a cosθ, b sinθ), parabola (at², 2at)

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 11 BoardHigh

JEE MainVery High

JEE AdvancedHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

All four curves (circle, ellipse, parabola, hyperbola) are obtained by cutting (sectioning) a CONE with a plane at different angles. Varying the angle of the cutting plane relative to the cone's axis produces different curves. This is why they are collectively called 'conic sections.'

The standard forms look almost identical — the only difference is a MINUS sign: ellipse x²/a²+y²/b²=1 vs hyperbola x²/a²−y²/b²=1. But the key difference is in c²: for ellipse c²=a²−b², for hyperbola c²=a²+b². The minus vs plus in c² is because ellipse has c<a while hyperbola has c>a.

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