Class 11 Maths Chapter 10: Conic Sections

Conic Sections is Chapter 10 of NCERT Class 11 Maths. It explains how slicing a double cone with a plane produces the circle, parabola, ellipse and hyperbola, and teaches the standard equations and key features of each curve. Watch the 4 video lessons below, then use the notes, formulas and solved examples to prepare for your CBSE exams.

What you will learn

How the four conic sections are formed from a double cone
The equation of a circle and how to find its centre and radius
Focus, directrix, axis and latus rectum of a parabola
Foci, vertices, axes and eccentricity of an ellipse and a hyperbola
How to read off every feature directly from a standard equation

Video Lessons

Conic Sections - Parabola

Conic Sections - Ellipse

Conic Sections - Circle

Conic Sections - Hyperbola

Key Concepts and Formulas

Sections of a Cone

Imagine a double cone formed by rotating a line (the generator) about a fixed vertical axis, keeping a constant angle α with the axis. When a plane cuts this cone at an angle β with the axis, the shape of the cut depends on how β compares with α.

β = 90°        → circle
α < β < 90°    → ellipse
β = α          → parabola
0 ≤ β < α      → hyperbola

If the plane passes through the vertex, we get degenerate conics: a point, a straight line or a pair of intersecting lines

Circle: Centre and Radius

A circle is the set of all points in a plane that are at a fixed distance (radius) from a fixed point (centre).

Centre (h, k), radius r:
(x − h)² + (y − k)² = r²
Centre at origin: x² + y² = r²
General form: x² + y² + 2gx + 2fy + c = 0
Centre = (−g, −f)
Radius = √(g² + f² − c)

To find the centre and radius from an expanded equation, complete the square in x and in y

Parabola: Focus and Directrix

A parabola is the set of all points that are equally distant from a fixed point (the focus) and a fixed line (the directrix). The line through the focus perpendicular to the directrix is the axis, and the point where the parabola meets its axis is the vertex.

y² = 4ax:  focus (a, 0),  directrix x = −a
y² = −4ax: focus (−a, 0), directrix x = a
x² = 4ay:  focus (0, a),  directrix y = −a
x² = −4ay: focus (0, −a), directrix y = a
Latus rectum = 4a  (a > 0)

Parabola y² = 20x with focus (5, 0), directrix x = −5 and latus rectum, Class 11 Maths Chapter 10 Conic Sections — PS equals PM for every point on the parabola

y² = 4ax opens right, y² = −4ax opens left, x² = 4ay opens upward, x² = −4ay opens downward
Tip: the variable that is squared tells you the axis of symmetry is the other axis

Ellipse: Foci and Eccentricity

An ellipse is the set of all points for which the sum of the distances from two fixed points (the foci) is constant. That constant equals the length of the major axis, 2a.

Major axis on x-axis: x²/a² + y²/b² = 1
Major axis on y-axis: x²/b² + y²/a² = 1
(a > b in both)
c² = a² − b²
Foci: (±c, 0) or (0, ±c)
Major axis = 2a, minor axis = 2b
e = c/a,  0 < e < 1
Latus rectum = 2b²/a

Ellipse x²/49 + y²/24 = 1 with foci (±5, 0) showing the sum of focal distances, Class 11 Conic Sections — Distances to the two foci always add up to 2a

Hyperbola: Transverse and Conjugate Axes

A hyperbola is the set of all points for which the difference of the distances from two fixed points (the foci) is constant. That constant equals 2a, the length of the transverse axis.

Transverse axis on x-axis: x²/a² − y²/b² = 1
Transverse axis on y-axis: y²/a² − x²/b² = 1
c² = a² + b²
Foci: (±c, 0) or (0, ±c)
Transverse axis = 2a, conjugate axis = 2b
e = c/a,  e > 1
Latus rectum = 2b²/a

Hyperbola x²/64 − y²/36 = 1 with foci (±10, 0) and asymptotes, showing difference of distances, Class 11 Conic Sections — Distances to the foci always differ by 2a

Unlike an ellipse, a can be smaller than b; the positive squared term decides the transverse axis

Solved Examples

Example 1: Find the equation of the circle with centre (−3, 2) and radius 5.

(x + 3)² + (y − 2)² = 5²
x² + 6x + 9 + y² − 4y + 4 = 25
∴ x² + y² + 6x − 4y − 12 = 0

Example 2: Find the centre and radius of the circle x² + y² − 10x + 6y + 18 = 0.

(x² − 10x) + (y² + 6y) = −18
(x − 5)² + (y + 3)² = −18 + 25 + 9
(x − 5)² + (y + 3)² = 16
∴ Centre (5, −3), radius 4

Example 3: For the parabola y² = 20x, find the focus, directrix and length of latus rectum.

4a = 20, so a = 5
Focus = (a, 0) = (5, 0)
Directrix: x = −a = −5
Latus rectum = 4a = 20
∴ Focus (5, 0), x = −5, latus rectum 20

Example 4: For the parabola x² = −24y, find the focus and the directrix.

Compare with x² = −4ay: 4a = 24, a = 6
Opens downward
Focus = (0, −a) = (0, −6)
∴ Focus (0, −6), directrix y = 6

Example 5: For the ellipse x²/49 + y²/24 = 1, find the foci, eccentricity, major axis and latus rectum.

a² = 49, b² = 24, so a = 7
c² = a² − b² = 49 − 24 = 25, c = 5
Foci = (±5, 0)
e = c/a = 5/7
Major axis = 2a = 14
Latus rectum = 2b²/a = 2(24)/7
∴ Foci (±5, 0), e = 5/7, LR = 48/7

Example 6: Find the equation of the ellipse with vertices (±10, 0) and foci (±6, 0).

a = 10, c = 6
b² = a² − c² = 100 − 36 = 64
∴ x²/100 + y²/64 = 1

Example 7: For the hyperbola x²/64 − y²/36 = 1, find the foci, eccentricity and latus rectum.

a² = 64, b² = 36, so a = 8
c² = a² + b² = 64 + 36 = 100, c = 10
Foci = (±10, 0)
e = c/a = 10/8 = 5/4
Latus rectum = 2b²/a = 2(36)/8 = 9
∴ Foci (±10, 0), e = 5/4, LR = 9

Common Mistakes to Avoid

Using c² = a² − b² for a hyperbola. For a hyperbola it is c² = a² + b²; the minus sign is only for an ellipse.
In an ellipse, assuming a² is always under x². The larger denominator is a² and decides the direction of the major axis.
Forgetting to change signs when writing the centre of a circle: (x + 3)² means h = −3, not 3.
Writing the latus rectum of a parabola as 2a instead of 4a.

Quick Revision

Circle: (x − h)² + (y − k)² = r²
General circle: centre (−g, −f)
r = √(g² + f² − c)
Parabola y² = 4ax: focus (a, 0), x = −a
Parabola latus rectum = 4a
Ellipse: c² = a² − b², e = c/a < 1
Hyperbola: c² = a² + b², e = c/a > 1
Ellipse, hyperbola latus rectum = 2b²/a