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Chapter 10: Conic Sections

Class 11 • Mathematics • Chapter 10

Conic Sections

The circle, parabola, ellipse and hyperbola, all born from slicing a cone.

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Chapter Roadmap

Sections of a Cone • Circle • Parabola • Ellipse • Hyperbola • Eccentricity • Key Results • Applications

Why Conic Sections Matter

Take a double cone and slice it with a flat plane. Depending on the angle of the cut, the edge of the slice is a circle, an ellipse, a parabola or a hyperbola. These four curves, the conic sections, appear all through science and engineering: planets move in ellipses, a thrown ball traces a parabola, and the shadow of a lamp on a wall can be a hyperbola.

In this chapter we study each conic through its equation in the coordinate plane. We will find the centre and radius of a circle, the focus and directrix of a parabola, and the foci and eccentricity of an ellipse and a hyperbola. A single number, the eccentricity, ties all four curves together and tells you which conic you are looking at, so it is worth keeping in mind from the start.

Key idea

All four conics come from cutting a cone. The eccentricity e decides which one you get: a circle has e = 0, an ellipse has e < 1, a parabola has e = 1, and a hyperbola has e > 1.

Key Terms You Must Know

Term

Meaning

Example

Conic section

A curve formed when a plane cuts a cone: circle, ellipse, parabola or hyperbola.

A planet’s orbit is an ellipse

Circle

All points at a fixed distance (radius) from a fixed point (centre).

(x − h)² + (y − k)² = r²

Focus

A fixed point used to define a conic.

Parabola y² = 4ax has focus (a, 0)

Directrix

A fixed line used with the focus to define a conic.

x = −a for y² = 4ax

Eccentricity

The number e that fixes the shape of a conic.

Ellipse e = c/a < 1

Latus rectum

The chord through a focus, perpendicular to the axis.

Length 4a for y² = 4ax

Vertex

A point where the curve meets its axis.

Parabola y² = 4ax has vertex (0, 0)

Core Concepts, Step by Step

1. Sections of a Cone

Imagine a double cone, two cones joined tip to tip. When a plane cuts straight across, parallel to the base, the section is a circle. Tilt the plane a little and the circle stretches into an ellipse. Tilt it until it is parallel to one slanting edge of the cone and the curve opens out into a parabola. Tilt it steeper still, so it cuts both halves of the double cone, and you get a hyperbola with its two branches. The same cone produces all four curves, which is why they are studied together.

2. The Circle

A circle is the set of all points that are a fixed distance, the radius r, from a fixed point, the centre (h, k). Using the distance formula, its equation is (x − h)² + (y − k)² = r². When the centre is the origin this becomes x² + y² = r². The same circle can also appear in the expanded general form x² + y² + 2gx + 2fy + c = 0, whose centre is (−g, −f) and whose radius is √(g² + f² − c).

The circle: centre C(h, k) and radius r

Circle with centre and radius on axes

3. The Parabola

A parabola is the set of all points equally far from a fixed point, the focus, and a fixed line, the directrix. Its standard equation, opening to the right, is y² = 4ax, with vertex at the origin, focus at (a, 0) and directrix x = −a. The chord through the focus perpendicular to the axis, the latus rectum, has length 4a. Similar equations describe parabolas opening left (y² = −4ax), upward (x² = 4ay) and downward (x² = −4ay).

The parabola y² = 4ax: vertex, focus and directrix

Parabola opening right with focus and directrix

4. The Ellipse

An ellipse is the set of all points whose distances from two fixed points, the foci, add up to a constant. Its standard equation is x²/a² + y²/b² = 1 with a > b. The foci sit at (±c, 0) where c² = a² − b², the length 2a is the major axis and 2b the minor axis. The eccentricity e = c/a lies between 0 and 1, and the latus rectum has length 2b²/a. A small e gives a nearly circular ellipse; a larger e gives a flatter one.

The ellipse: two foci, major and minor axes

Ellipse with foci and axes

5. The Hyperbola

A hyperbola is the set of all points whose distances from two foci have a constant difference. Its standard equation is x²/a² − y²/b² = 1, giving two opening branches. The foci are at (±c, 0) where this time c² = a² + b², so the only change from the ellipse is a plus sign. The eccentricity e = c/a is greater than 1, and the latus rectum again has length 2b²/a. Take care over that single sign: it is the most common slip in this chapter.

The hyperbola: two branches, foci and asymptotes

Hyperbola with two branches, foci and asymptotes

6. Eccentricity Ties Them Together

Although the four conics look different, one number unites them. The eccentricity e measures how far a conic departs from being a circle. A circle has e = 0, an ellipse has 0 < e < 1, a parabola has e = 1 exactly, and a hyperbola has e > 1. The table below collects the standard equations and key features so you can compare them at a glance.

The four conics at a glance

Conic

Standard equation

Foci relation

Eccentricity

Circle

x² + y² = r²

centre and radius r

e = 0

Parabola

y² = 4ax

focus (a, 0)

e = 1

Ellipse

x²/a² + y²/b² = 1

c² = a² − b²

e = c/a < 1

Hyperbola

x²/a² − y²/b² = 1

c² = a² + b²

e = c/a > 1

Key Results & Proofs

Three results give the equations their meaning. Each comes directly from the distance formula and the definition of the curve.

Result 1: The Equation of a Circle

Statement. The circle with centre (h, k) and radius r has equation (x − h)² + (y − k)² = r².

Proof

Apply the distance formula to the centre and a general point.

A point (x, y) is on the circle when its distance from the centre is r. – definition of a circle

√[(x − h)² + (y − k)²]

=

r

distance from (x, y) to (h, k)

(x − h)² + (y − k)²

=

r²

square both sides

With centre at the origin (h = 0, k = 0) this reduces to the familiar x² + y² = r².

Result 2: The Standard Parabola

Statement. A parabola with focus (a, 0) and directrix x = −a has equation y² = 4ax.

Proof

Set the distance to the focus equal to the distance to the directrix.

A point (x, y) is on the parabola when its distance from the focus equals its distance from the directrix. – definition of a parabola

√[(x − a)² + y²]

=

x + a

distance to focus equals distance to directrix

(x − a)² + y²

=

(x + a)²

square both sides

y²

=

(x + a)² − (x − a)²

move the x term across

y²

=

4ax

expand and simplify

Putting x = a gives y² = 4a², so y = ±2a; the latus rectum has length 4a.

Result 3: Centre and Radius From the General Form

Statement. The circle x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).

Proof

Complete the square to turn the general form into the standard form.

x² + 2gx + y² + 2fy

=

−c

group the x and y terms

(x + g)² + (y + f)²

=

g² + f² − c

complete the square on each pair

Comparing with (x − h)² + (y − k)² = r²: – match the standard form

centre is (−g, −f) and radius is √(g² + f² − c). – read off the answer

If g² + f² − c is negative there is no real circle, since a radius cannot be imaginary.

Worked Examples

Example 1

Question: Find the centre and radius of the circle (x − 2)² + (y + 3)² = 25.

▶ Show full working

Compare with (x − h)² + (y − k)² = r².

Matching gives h = 2, k = −3 and r² = 25. – read off the values

So the centre is (2, −3) and r = √25 = 5. – take the square root

Answer: Centre (2, −3), radius 5.

Example 2

Question: Find the equation of the circle with centre (1, 2) and radius 3.

▶ Show full working

Substitute into (x − h)² + (y − k)² = r².

(x − 1)² + (y − 2)²

=

3²

substitute the centre and radius

(x − 1)² + (y − 2)²

=

9

simplify

Answer: (x − 1)² + (y − 2)² = 9.

Example 3

Question: Find the centre and radius of x² + y² − 4x − 6y − 12 = 0.

▶ Show full working

Compare with x² + y² + 2gx + 2fy + c = 0.

Here 2g = −4, 2f = −6, c = −12, so g = −2, f = −3. – read off g, f, c

Centre = (−g, −f) = (2, 3). – centre formula

r

=

√(g² + f² − c) = √(4 + 9 + 12)

radius formula

r

=

√25 = 5

simplify

Answer: Centre (2, 3), radius 5.

Example 4

Question: For the parabola y² = 12x, find the focus, directrix and length of the latus rectum.

▶ Show full working

Compare with y² = 4ax to find a.

4a = 12, so a = 3. – compare with y² = 4ax

Focus is (a, 0) = (3, 0). – focus formula

Directrix is x = −a, that is x = −3. – directrix formula

Latus rectum = 4a = 12. – latus rectum formula

Answer: Focus (3, 0), directrix x = −3, latus rectum 12.

Example 5

Question: Find the equation of the parabola with focus (2, 0) and directrix x = −2.

▶ Show full working

The focus is (a, 0) and the directrix x = −a, so read off a.

Here a = 2, so 4a = 8. – compare with the standard parabola

y²

=

4ax = 8x

substitute

Answer: y² = 8x.

Example 6

Question: For the ellipse x²/25 + y²/9 = 1, find a, b, the foci and the eccentricity.

▶ Show full working

Compare with x²/a² + y²/b² = 1, then use c² = a² − b².

a² = 25 and b² = 9, so a = 5, b = 3. – read off a and b

c

=

√(a² − b²) = √(25 − 9) = 4

find c

Foci are (±c, 0) = (±4, 0). – foci formula

e

=

c/a = 4/5

eccentricity

Answer: a = 5, b = 3, foci (±4, 0), e = 4/5.

Example 7

Question: Find the length of the latus rectum of the ellipse x²/16 + y²/9 = 1.

▶ Show full working

Use latus rectum = 2b²/a, with a the larger semi-axis.

a² = 16 and b² = 9, so a = 4. – identify a and b

latus rectum

=

2b²/a = 2(9)/4

apply the formula

latus rectum

=

18/4 = 4.5

simplify

Answer: The latus rectum is 4.5 units.

Example 8

Question: For the hyperbola x²/9 − y²/16 = 1, find the foci and the eccentricity.

▶ Show full working

Compare with x²/a² − y²/b² = 1, then use c² = a² + b².

a² = 9, b² = 16, so a = 3, b = 4. – read off a and b

c

=

√(a² + b²) = √(9 + 16) = 5

note the plus sign

Foci are (±5, 0). – foci formula

e

=

c/a = 5/3

eccentricity, greater than 1

Answer: Foci (±5, 0), eccentricity 5/3.

Example 9

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

▶ Show full working

Use c = 3, a = 5 and b² = a² − c².

b²

=

a² − c² = 25 − 9

find b²

b²

=

16

simplify

x²/25 + y²/16

=

1

write the equation

Answer: x²/25 + y²/16 = 1.

Example 10

Question: Find the equation of the circle with centre (3, 4) that passes through the origin.

▶ Show full working

The radius is the distance from the centre to the origin.

r

=

√[(3 − 0)² + (4 − 0)²]

distance from (3, 4) to (0, 0)

r

=

√(9 + 16) = 5

simplify

(x − 3)² + (y − 4)²

=

25

standard form with r = 5

Answer: (x − 3)² + (y − 4)² = 25.

Example 11

Question: For the parabola x² = 8y, find the focus and the directrix.

▶ Show full working

This parabola opens upward; compare with x² = 4ay.

4a = 8, so a = 2. – compare with x² = 4ay

Focus is (0, a) = (0, 2). – focus on the y-axis

Directrix is y = −a, that is y = −2. – directrix below the vertex

Answer: Focus (0, 2), directrix y = −2.

Example 12

Question: Find the eccentricity of the hyperbola x²/16 − y²/9 = 1.

▶ Show full working

Find c using c² = a² + b², then e = c/a.

a² = 16, b² = 9, so a = 4. – read off a and b

c

=

√(16 + 9) = 5

find c

e

=

c/a = 5/4

eccentricity

Answer: The eccentricity is 5/4.

Where You Meet This in Real Life

Planetary orbits

Planets and comets move around the Sun in ellipses, with the Sun at one focus, a discovery made by Kepler.

Projectiles

A ball, a jet of water or a long jump follows a parabolic path under gravity, ignoring air resistance.

Dishes and headlights

Satellite dishes, telescopes and car headlights use parabolic shapes to focus or spread out signals and light.

Bridges and arches

Suspension cables and many arches take the shape of a parabola, which spreads load smoothly.

Navigation and design

Hyperbolas appear in long-range navigation systems and in the cooling towers of power stations, whose curved sides are hyperbolic.

Practice Sets A to D

Practice Set A – Basics

A1. Find the centre and radius of (x + 1)² + (y − 2)² = 16.

▶ Reveal full working

Compare with the standard form.

h = −1, k = 2, r² = 16. – read off the values

Centre (−1, 2), radius 4. – take the square root

Answer: Centre (−1, 2), radius 4.

A2. Write the equation of the circle with centre at the origin and radius 7.

▶ Reveal full working

Use x² + y² = r².

x² + y²

=

49

substitute r = 7

Answer: x² + y² = 49.

A3. For the parabola y² = 4x, find the focus.

▶ Reveal full working

Compare with y² = 4ax.

4a = 4, so a = 1. – find a

Focus is (a, 0) = (1, 0). – focus formula

Answer: Focus (1, 0).

A4. Is x²/9 + y²/4 = 1 an ellipse or a hyperbola?

▶ Reveal full working

Look at the sign between the two terms.

The two terms are added, so it is an ellipse. – plus sign means ellipse

Answer: An ellipse.

Practice Set B – Conceptual

B1. What shape is formed when a plane cuts a cone parallel to its base?

▶ Reveal full working

Think about the slice across the cone.

A cut parallel to the base gives a perfectly round slice.

That slice is a circle.

Answer: A circle.

B2. What is the eccentricity of a circle, and of a parabola?

▶ Reveal full working

Recall the eccentricity scale.

A circle does not depart from roundness, so e = 0.

A parabola sits exactly at the boundary, so e = 1.

Answer: Circle: e = 0; parabola: e = 1.

B3. How does the foci relation differ between an ellipse and a hyperbola?

▶ Reveal full working

Compare the two formulae for c.

For an ellipse, c² = a² − b².

For a hyperbola, c² = a² + b².

Answer: Ellipse uses a minus; hyperbola uses a plus.

B4. What is the latus rectum of a conic?

▶ Reveal full working

Describe the chord through the focus.

It is the chord that passes through a focus.

It is drawn perpendicular to the axis of the conic.

Answer: The chord through a focus, perpendicular to the axis.

Practice Set C – Application / Numerical

C1. Find the centre and radius of x² + y² − 6x + 4y − 3 = 0.

▶ Reveal full working

Compare with the general form.

2g = −6, 2f = 4, c = −3, so g = −3, f = 2. – read off g, f, c

Centre = (3, −2). – centre formula

r

=

√(9 + 4 + 3) = √16 = 4

radius formula

Answer: Centre (3, −2), radius 4.

C2. For the parabola y² = 20x, find the latus rectum and the directrix.

▶ Reveal full working

Compare with y² = 4ax.

4a = 20, so a = 5. – find a

Latus rectum = 4a = 20. – latus rectum

Directrix x = −a = −5. – directrix

Answer: Latus rectum 20, directrix x = −5.

C3. Find the eccentricity of the ellipse x²/100 + y²/64 = 1.

▶ Reveal full working

Use c² = a² − b² and e = c/a.

a = 10, b = 8. – read off a and b

c

=

√(100 − 64) = 6

find c

e

=

c/a = 6/10 = 0.6

eccentricity

Answer: e = 0.6.

C4. Find the foci of the hyperbola x²/25 − y²/144 = 1.

▶ Reveal full working

Use c² = a² + b².

a² = 25, b² = 144. – read off the values

c

=

√(25 + 144) = √169 = 13

find c

Foci are (±13, 0). – foci formula

Answer: Foci (±13, 0).

Practice Set D – HOTS / Word Problems

D1. Find the equation of the ellipse with vertices (±6, 0) and foci (±4, 0).

▶ Reveal full working

Read a and c, then find b² = a² − c².

a = 6 (vertices) and c = 4 (foci). – read off a and c

b²

=

a² − c² = 36 − 16 = 20

find b²

x²/36 + y²/20

=

1

write the equation

Answer: x²/36 + y²/20 = 1.

D2. Find the equation of the parabola with vertex at the origin, axis along the x-axis, passing through (2, 3).

▶ Reveal full working

Use y² = 4ax and substitute the point to find a.

y²

=

4ax

standard parabola

3²

=

4a(2)

substitute (2, 3)

9

=

8a, so a = 9/8

solve for a

y²

=

(9/2)x

substitute 4a = 9/2

Answer: y² = (9/2)x.

D3. A circle has centre (2, −3) and passes through (5, 1). Find its equation.

▶ Reveal full working

The radius is the distance from the centre to the given point.

r

=

√[(5 − 2)² + (1 + 3)²]

distance from centre to point

r

=

√(9 + 16) = 5

simplify

(x − 2)² + (y + 3)²

=

25

standard form

Answer: (x − 2)² + (y + 3)² = 25.

D4. Find the eccentricity of an ellipse whose latus rectum is half its major axis.

▶ Reveal full working

Set 2b²/a equal to half of 2a, then use e² = 1 − b²/a².

2b²/a

=

a

latus rectum is half the major axis 2a

2b²

=

a², so b²/a² = 1/2

rearrange

e²

=

1 − b²/a² = 1 − 1/2 = 1/2

eccentricity relation

e

=

1/√2

take the square root

Answer: e = 1/√2.

Chapter Summary

Everything in One Glance

Sections of a Cone

Cutting a cone gives a circle, ellipse, parabola or hyperbola, depending on the angle of the cut.

Circle

(x − h)² + (y − k)² = r²; general form x² + y² + 2gx + 2fy + c = 0.

Parabola

y² = 4ax: vertex (0,0), focus (a,0), directrix x = −a, latus rectum 4a.

Ellipse

x²/a² + y²/b² = 1: c² = a² − b², foci (±c,0), e = c/a < 1.

Hyperbola

x²/a² − y²/b² = 1: c² = a² + b², foci (±c,0), e = c/a > 1.

Eccentricity

Circle e = 0, ellipse e < 1, parabola e = 1, hyperbola e > 1.

Are You Exam-Ready?

8-Point Exam Quick-Check

Find the centre and radius of (x − 3)² + (y + 1)² = 49.

Write the circle with centre origin and radius 6.

For y² = 16x, find the focus and the latus rectum.

For x²/36 + y²/11 = 1, find the foci.

For x²/4 − y²/5 = 1, find the eccentricity.

State the eccentricity of a circle, an ellipse, a parabola and a hyperbola.

Find the centre and radius of x² + y² − 2x − 8 = 0.

Find the equation of the parabola with focus (0, 3) opening upward.

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Class 11 Mathematics Chapter 10: Conic Sections, Complete Notes and Practice

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