Conic Sections – Class 11 Maths Chapter 10 Notes & NCERT PDF

Class 11MathematicsChapter 10NCERT book: Mathematics - Class 11

Notes and a simple summary of Chapter 10 of the NCERT Class 11 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 10: Conic Sections

Introduction

Chapter 10, Conic Sections, begins with Bertrand Russell's advice to let "the relation of knowledge to real life be very visible." Circles, parabolas, ellipses and hyperbolas are the curves you get by slicing a double-napped cone with a plane. They describe planetary orbits, the paths of projectiles, the reflectors in torches and car headlights, satellite dishes and the cooling towers of power stations. In this chapter each conic is defined as a locus of points and turned into a standard equation.

Key Concepts & Theorems

10.2 Sections of a Cone

A double-napped right circular cone is generated when a line (the generator) rotates about a fixed line (the axis) through a fixed point (the vertex), keeping a constant angle α with the axis. If a plane makes angle β with the axis and does not pass through the vertex:

β = 90° → circle

α < β < 90° → ellipse

β = α → parabola

0 ≤ β < α → hyperbola (the plane cuts both nappes)

When the plane passes through the vertex we get degenerate conics (10.2.2): a point, a pair of intersecting lines or a single line.

10.3 Circle

A circle is the set of all points in a plane equidistant from a fixed point (centre). With centre (h, k) and radius r:

(x − h)² + (y − k)² = r²

With centre at the origin: x² + y² = r².

10.4 Parabola

A parabola is the set of all points in a plane equidistant from a fixed line (directrix) and a fixed point (focus) not on the line. The line through the focus perpendicular to the directrix is the axis; the point where the parabola meets the axis is the vertex.

The four standard equations (vertex at origin, a > 0):

y² = 4ax: focus (a, 0), directrix x = −a, opens right

y² = −4ax: focus (−a, 0), directrix x = a, opens left

x² = 4ay: focus (0, a), directrix y = −a, opens upward

x² = −4ay: focus (0, −a), directrix y = a, opens downward

Latus rectum (10.4.2): the chord through the focus perpendicular to the axis; its length is 4a.

10.5 Ellipse

An ellipse is the set of all points in a plane the sum of whose distances from two fixed points (foci) is constant (= 2a). The major axis (length 2a) joins the vertices through the foci; the minor axis has length 2b; c is the distance of each focus from the centre.

Relation (10.5.1): a² = b² + c², i.e. c = √(a² − b²).

Eccentricity (10.5.2): e = c/a, and 0 < e < 1 for an ellipse.

Standard equations (10.5.3): x²/a² + y²/b² = 1 (foci (±c, 0), major axis along x-axis) and x²/b² + y²/a² = 1 (foci (0, ±c), major axis along y-axis), with a > b.

Latus rectum (10.5.4): length 2b²/a.

10.6 Hyperbola

A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points (foci) is constant (= 2a). The transverse axis passes through the foci; the conjugate axis is perpendicular to it through the centre.

Relation: c² = a² + b².

Eccentricity (10.6.1): e = c/a > 1.

Standard equations (10.6.2): x²/a² − y²/b² = 1 (foci (±c, 0)) and y²/a² − x²/b² = 1 (foci (0, ±c)).

Latus rectum (10.6.3): length 2b²/a.

Worked Examples

Example 2 & 3 (circle). Centre (−3, 2), radius 4: (x + 3)² + (y − 2)² = 16. For x² + y² + 8x + 10y − 8 = 0, complete squares: (x + 4)² + (y + 5)² = 8 + 16 + 25 = 49, so centre (−4, −5), radius 7.

Example 4 (circle through two points). The circle passes through (2, −2) and (3, 4) and its centre lies on x + y = 2. Equating distances and using h + k = 2 gives h = 0.7, k = 1.3, r² = 12.58: (x − 0.7)² + (y − 1.3)² = 12.58.

Example 5 (parabola). For y² = 8x, 4a = 8 ⇒ a = 2: focus (2, 0), axis the x-axis, directrix x = −2, latus rectum 8.

Example 8 (parabola through a point). Symmetric about the y-axis and passing through (2, −3), which is below the x-axis, so x² = −4ay. Then 4 = 12a ⇒ a = 1/3, giving 3x² = −4y.

Example 9 (ellipse). For x²/25 + y²/9 = 1: a = 5, b = 3, c = √(25 − 9) = 4. Foci (±4, 0), vertices (±5, 0), major axis 10, minor axis 6, e = 4/5, latus rectum 2b²/a = 18/5.

Example 11 (ellipse from data). Vertices (±13, 0), foci (±5, 0): b² = 169 − 25 = 144, so x²/169 + y²/144 = 1.

Example 14 (hyperbola). For x²/9 − y²/16 = 1: a = 3, b = 4, c = √(9 + 16) = 5; foci (±5, 0), vertices (±3, 0), e = 5/3, latus rectum 32/3. For y² − 16x² = 16, i.e. y²/16 − x²/1 = 1: a = 4, b = 1, c = √17; foci (0, ±√17), vertices (0, ±4), e = √17/4, latus rectum 1/2.

Example 17 (application). A parabolic mirror has its focus 5 cm from the vertex and is 45 cm deep. With y² = 20x and x = 45, y² = 900, y = ±30, so the width AB = 60 cm.

Formula Sheet

Circle: (x − h)² + (y − k)² = r²

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

Ellipse x²/a² + y²/b² = 1 (a > b): c² = a² − b², e = c/a < 1, latus rectum 2b²/a

Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a > 1, latus rectum 2b²/a

Ellipse/hyperbola with foci on the y-axis: swap the roles of x and y

Exam Focus

Exercises 10.1–10.4 ask for (i) the centre and radius from a general circle equation, (ii) focus, axis, directrix and latus rectum from a parabola's equation, (iii) foci, vertices, axes, eccentricity and latus rectum of an ellipse or hyperbola, and (iv) the equation of a conic from given data.

Outline — "Find foci, vertices, e and latus rectum of an ellipse": decide which denominator is larger (that variable carries the major axis) → read a² and b² → c = √(a² − b²) → list foci, vertices, 2a, 2b, e = c/a, 2b²/a.

Outline — "Find the equation of a parabola given focus and directrix": place the focus and directrix, decide the direction of opening, read a, write the matching standard form.

Outline — "Find the equation of a hyperbola given foci and vertices": foci give c and orientation, vertices give a, then b² = c² − a².

Key Terms

Conic section — शंकु परिच्छेद

Circle — वृत्त

Parabola — परवलय

Ellipse — दीर्घवृत्त

Hyperbola — अतिपरवलय

Focus (foci) — नाभि (नाभियाँ)

Directrix — नियता

Eccentricity — उत्केंद्रता

Latus rectum — नाभिलंब

Vertex — शीर्ष

Major / minor axis — दीर्घ अक्ष / लघु अक्ष

Transverse / conjugate axis — अनुप्रस्थ अक्ष / संयुग्मी अक्ष

Locus — बिंदुपथ

Nappe (of a cone) — शंकु का भाग (नेप)

Common Mistakes

Using c² = a² − b² for a hyperbola (it is c² = a² + b²) — and vice versa for the ellipse.

Assuming the x² term always carries a² in an ellipse: the larger denominator decides the major axis.

Forgetting the sign when a parabola opens left or down (y² = −4ax, x² = −4ay).

Writing the latus rectum of a parabola as 2a instead of 4a.

Stating an eccentricity greater than 1 for an ellipse or less than 1 for a hyperbola.

Completing the square incorrectly in a circle's general equation (add the squares of half the coefficients to the constant side).

💡 Key Learning Points

✓Circles, ellipses, parabolas and hyperbolas arise when a plane cuts a double-napped cone at different angles; planes through the vertex give degenerate conics.
✓Circle: (x − h)² + (y − k)² = r²; completing the square gives the centre and radius from a general equation.
✓Parabola y² = 4ax has focus (a, 0), directrix x = −a and latus rectum 4a; the other three standard forms follow by symmetry.
✓Ellipse x²/a² + y²/b² = 1 (a > b) has c² = a² − b², e = c/a < 1 and latus rectum 2b²/a.
✓Hyperbola x²/a² − y²/b² = 1 has c² = a² + b², e = c/a > 1 and latus rectum 2b²/a.

👨‍🏫 Teaching Tips

→Shine a torch on a wall at different tilts to show circle, ellipse, parabola and hyperbola as light sections of a cone.
→Draw an ellipse with two pins and a loop of string to make 'sum of distances is constant' concrete.
→Use a comparison chart (as a list) of the four standard parabolas with sketches showing direction of opening.
→Stress the one-line test: the larger denominator decides the major axis of an ellipse.
→Contrast ellipse and hyperbola side by side: c² = a² − b² vs c² = a² + b², e < 1 vs e > 1.
→Link Example 17 (parabolic mirror) to satellite dishes and car headlights.
→Give students general circle equations to convert to centre–radius form as a warm-up drill.

📋 Assessment Questions

Find the centre and radius of x² + y² + 8x + 10y − 8 = 0.
Find the focus, directrix and length of latus rectum of y² = 8x.
Find the foci, vertices, eccentricity and latus rectum of x²/25 + y²/9 = 1.
Find the equation of the ellipse with vertices (±13, 0) and foci (±5, 0).
Find the foci and eccentricity of the hyperbola x²/9 − y²/16 = 1.

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