Chapter Introduction
Conic Sections is Chapter 10 of CBSE Class 11 Mathematics, falling under the Coordinate Geometry unit. Conic sections are curves obtained by intersecting a double-napped right circular cone with a plane at different angles. Depending on the angle of the plane, we get a circle, ellipse, parabola, or hyperbola. This chapter typically carries 8-10 marks in board exams and is extremely important for JEE geometry.
The unifying idea: a conic section is the locus of a point that moves such that its distance from a fixed point (focus) bears a constant ratio (eccentricity e) to its distance from a fixed line (directrix).
Key Definitions
| Term | Definition |
|---|---|
| Conic Section | Locus of a point P such that PF/PD = e (constant), where F is focus and D is foot of perpendicular on directrix |
| Focus (F) | The fixed point used to define the conic |
| Directrix | The fixed line used to define the conic |
| Eccentricity (e) | The constant ratio PF/PD; determines the type of conic |
| Axis | The line through the focus perpendicular to the directrix |
| Vertex | The point where the conic intersects its axis |
| Latus Rectum | The chord through the focus, perpendicular to the axis |
| Centre | The midpoint of the line segment joining the two foci (for ellipse and hyperbola) |
| Major Axis | The longest diameter of an ellipse (length 2a) |
| Minor Axis | The shortest diameter of an ellipse (length 2b) |
| Transverse Axis | The line segment joining the two vertices of a hyperbola (length 2a) |
| Conjugate Axis | The line segment of length 2b perpendicular to the transverse axis through the centre |
| Asymptotes | Lines that a hyperbola approaches but never touches |
Detailed Concepts
1. Sections of a Cone
| Angle of Plane | Section Formed | Eccentricity |
|---|---|---|
| Perpendicular to axis | Circle | e = 0 |
| Tilted (less than generator angle) | Ellipse | 0 < e < 1 |
| Parallel to a generator | Parabola | e = 1 |
| Steeper than generator | Hyperbola | e > 1 |
| Through vertex (special cases) | Point, line, or two intersecting lines (degenerate conics) | — |
2. Circle
Standard equation (centre at origin): x^2 + y^2 = r^2
General equation (centre at (h, k)): (x - h)^2 + (y - k)^2 = r^2
General expanded form: x^2 + y^2 + 2gx + 2fy + c = 0, where centre = (-g, -f) and radius = sqrt(g^2 + f^2 - c)
For a circle: eccentricity e = 0, no directrix, focus = centre.
3. Parabola
Standard form (axis along x-axis, opening right): y^2 = 4ax
| Property | Value |
|---|---|
| Vertex | (0, 0) |
| Focus | (a, 0) |
| Directrix | x = -a |
| Axis | x-axis (y = 0) |
| Latus rectum length | 4a |
| Latus rectum endpoints | (a, 2a) and (a, -2a) |
| Eccentricity | e = 1 |
Other standard forms of parabola:
| Equation | Opening Direction | Focus | Directrix |
|---|---|---|---|
| y^2 = 4ax | Right (along +x) | (a, 0) | x = -a |
| y^2 = -4ax | Left (along -x) | (-a, 0) | x = a |
| x^2 = 4ay | Up (along +y) | (0, a) | y = -a |
| x^2 = -4ay | Down (along -y) | (0, -a) | y = a |
4. Ellipse
Standard form (major axis along x-axis): x^2/a^2 + y^2/b^2 = 1, where a > b
| Property | Value (a > b) | Value (b > a) |
|---|---|---|
| Foci | (+/-ae, 0) | (0, +/-be) |
| Vertices | (+/-a, 0) | (0, +/-b) |
| Centre | (0, 0) | (0, 0) |
| Major axis length | 2a | 2b |
| Minor axis length | 2b | 2a |
| Eccentricity | e = sqrt(1 - b^2/a^2) | e = sqrt(1 - a^2/b^2) |
| Directrices | x = +/-a/e | y = +/-b/e |
| Latus rectum length | 2b^2/a | 2a^2/b |
Key relationship: c^2 = a^2 - b^2 (where c = ae is the distance from centre to focus)
5. Hyperbola
Standard form (transverse axis along x-axis): x^2/a^2 - y^2/b^2 = 1
| Property | Value |
|---|---|
| Foci | (+/-ae, 0) |
| Vertices | (+/-a, 0) |
| Centre | (0, 0) |
| Transverse axis length | 2a |
| Conjugate axis length | 2b |
| Eccentricity | e = sqrt(1 + b^2/a^2) |
| Directrices | x = +/-a/e |
| Latus rectum length | 2b^2/a |
| Asymptotes | y = +/-(b/a)x |
Key relationship: c^2 = a^2 + b^2 (note the plus sign, unlike ellipse)
Rectangular hyperbola: When a = b, equation is x^2 - y^2 = a^2, eccentricity e = sqrt(2), asymptotes are y = +/-x.
6. Degenerate Conics
When the plane passes through the vertex of the cone:
- Point conic: the vertex itself
- Line (straight line): the generator
- Two intersecting lines: two generators
Conic Comparison Table
| Property | Circle | Parabola | Ellipse | Hyperbola |
|---|---|---|---|---|
| Eccentricity | 0 | 1 | 0 < e < 1 | e > 1 |
| Focus | Centre | One focus | Two foci | Two foci |
| Directrix | None | One | Two | Two |
| Centre | Yes | No | Yes | Yes |
| Standard Equation | x^2 + y^2 = r^2 | y^2 = 4ax | x^2/a^2 + y^2/b^2 = 1 | x^2/a^2 - y^2/b^2 = 1 |
| Latus Rectum | 2r | 4a | 2b^2/a | 2b^2/a |
| Asymptotes | None | None | None | y = +/-(b/a)x |
| c relation | — | — | c^2 = a^2 - b^2 | c^2 = a^2 + b^2 |
Important Numericals
Example 1: Finding vertex, focus, directrix of parabola
For y^2 = 12x, identify all elements.
Comparing with y^2 = 4ax: 4a = 12, so a = 3.
Vertex: (0, 0), Focus: (3, 0), Directrix: x = -3, Axis: x-axis, Latus rectum = 12, Endpoints: (3, 6) and (3, -6).
Example 2: Ellipse elements
Find the foci, vertices, eccentricity, and latus rectum of x^2/25 + y^2/9 = 1.
a^2 = 25, b^2 = 9, a = 5, b = 3.
c = sqrt(25 - 9) = sqrt(16) = 4, e = c/a = 4/5.
Foci: (+/-4, 0), Vertices: (+/-5, 0), Latus rectum = 2(9)/5 = 18/5.
Example 3: Hyperbola elements
Find the foci, vertices, eccentricity, and asymptotes of x^2/16 - y^2/9 = 1.
a^2 = 16, b^2 = 9, a = 4, b = 3.
c = sqrt(16 + 9) = sqrt(25) = 5, e = c/a = 5/4.
Foci: (+/-5, 0), Vertices: (+/-4, 0), Latus rectum = 2(9)/4 = 9/2.
Asymptotes: y = +/-(3/4)x, i.e., 3x - 4y = 0 and 3x + 4y = 0.
Example 4: Finding equation from conditions
Find the equation of the ellipse with foci (+/-3, 0) and a = 5.
c = 3, a = 5. b^2 = a^2 - c^2 = 25 - 9 = 16, b = 4.
Equation: x^2/25 + y^2/16 = 1
Example 5: Parabola from focus and directrix
Find the equation of the parabola with focus (0, -3) and directrix y = 3.
Focus at (0, -3), directrix y = 3. Vertex is the midpoint: (0, 0).
a = distance from vertex to focus = 3. Since focus is below vertex and directrix is above, the parabola opens downward.
Equation: x^2 = -4ay = -4(3)y = -12y, i.e., x^2 = -12y.
Common Mistakes to Avoid
1. Confusing ellipse and hyperbola signs: x^2/a^2 + y^2/b^2 = 1 is an ellipse; x^2/a^2 - y^2/b^2 = 1 is a hyperbola. The sign of y^2 term is the key difference.
2. Wrong c relation: For ellipse, c^2 = a^2 - b^2 (minus). For hyperbola, c^2 = a^2 + b^2 (plus). This is the most common error.
3. Parabola opening direction: y^2 = 4ax opens right; y^2 = -4ax opens left; x^2 = 4ay opens up; x^2 = -4ay opens down. Check the sign carefully.
4. Eccentricity range: Circle: e = 0, Ellipse: 0 < e < 1, Parabola: e = 1, Hyperbola: e > 1. Eccentricity is never negative.
5. a and b in ellipse: a is always the semi-major axis (larger), b is the semi-minor axis. If the denominator under x^2 is smaller than under y^2, then the major axis is along y.
6. Latus rectum formula: For both ellipse and hyperbola, latus rectum = 2b^2/a. The formula is the same but b has different meaning.
7. Forgetting conditions: Always state whether a > b or b > a before writing foci and directrix for ellipse. The answer changes based on this.
Exam Tips
- For 1-mark questions, know: eccentricity values, standard forms, definitions of focus/directrix/latus rectum.
- For 3-mark questions, practice: finding elements from equation, writing equation from given conditions.
- For 5-mark questions, practice: deriving equations from focus-directrix definition, and problems requiring you to find the equation of a conic passing through given points.
- Always write the standard form you are comparing with before identifying a, b, c.
- For parabola problems, identify the axis first (is it x^2 = ... or y^2 = ...), then determine the opening direction.
- Draw a rough diagram for every problem — it helps avoid sign errors.
- In ellipse, always check whether a > b or b > a before writing foci positions.
NCERT Exercise Problem Patterns
| Exercise | Topic | Typical Question Type |
|---|---|---|
| 10.1 | Sections of cone, circle | Find equation of circle, centre and radius |
| 10.2 | Parabola | Find vertex, focus, directrix, latus rectum; equation from conditions |
| 10.3 | Ellipse | Find foci, vertices, eccentricity; equation from given data |
| 10.4 | Hyperbola | Find foci, vertices, eccentricity, asymptotes; equation from conditions |
Derivations to Remember
Derivation 1: Standard equation of parabola (focus-directrix definition)
Let focus F = (a, 0) and directrix x = -a. Let P(x, y) be a point on the parabola.
By definition: PF = PD (distance to focus = distance to directrix, since e = 1)
| sqrt((x - a)^2 + y^2) = | x + a |
|---|
Squaring: (x - a)^2 + y^2 = (x + a)^2
x^2 - 2ax + a^2 + y^2 = x^2 + 2ax + a^2
y^2 = 4ax
Derivation 2: Standard equation of ellipse
For ellipse, sum of distances from two foci is constant = 2a.
Foci: F1 = (ae, 0), F2 = (-ae, 0). Let P(x, y) be on the ellipse.
PF1 + PF2 = 2a
sqrt((x - ae)^2 + y^2) + sqrt((x + ae)^2 + y^2) = 2a
After squaring twice and simplifying (using c = ae and b^2 = a^2 - c^2):
x^2/a^2 + y^2/b^2 = 1
Derivation 3: Standard equation of hyperbola
For hyperbola, difference of distances from two foci is constant = 2a.
| PF1 - PF2 | = 2a |
|---|
Following similar algebra as ellipse, using c^2 = a^2 + b^2:
x^2/a^2 - y^2/b^2 = 1
Additional Solved Examples
Example 6: Circle from diameter endpoints
Find the equation of the circle with diameter endpoints A(1, 1) and B(4, 5).
Centre = midpoint = (5/2, 3). Radius = (1/2)distance AB = (1/2)sqrt(9 + 16) = (1/2)*5 = 5/2
Equation: (x - 5/2)^2 + (y - 3)^2 = 25/4
Expanding: x^2 - 5x + 25/4 + y^2 - 6y + 9 = 25/4
x^2 + y^2 - 5x - 6y + 9 = 0
Example 7: Parabola from vertex and focus
Find the equation of the parabola with vertex (2, 3) and focus (2, 5).
Focus is directly above vertex, so axis is vertical (x = 2) and parabola opens upward.
a = distance from vertex to focus = 2. Equation: (x - 2)^2 = 4(2)(y - 3) = 8(y - 3)
Example 8: Hyperbola with given asymptotes
If the asymptotes of a hyperbola are y = (3/4)x and y = -(3/4)x, find the equation if it passes through (4, 0).
b/a = 3/4, so b = 3k and a = 4k for some k.
Equation: x^2/(16k^2) - y^2/(9k^2) = 1. Point (4, 0): 16/(16k^2) = 1, so k = 1.
Equation: x^2/16 - y^2/9 = 1
Practice Checklist
- Identify all elements (vertex, focus, directrix, LR) for each conic type (at least 2 each)
- Write equations from given conditions (at least 3 problems)
- Solve 2 problems for each conic type: circle, parabola, ellipse, hyperbola
- Practice the focus-directrix derivation for parabola
- Draw a rough sketch for every problem to check signs and orientation
Quick Revision Notes
1. Circle: x^2 + y^2 = r^2; centre (0,0), radius r; e = 0 2. Parabola: y^2 = 4ax; focus (a,0), directrix x = -a, LR = 4a; e = 1 3. Ellipse: x^2/a^2 + y^2/b^2 = 1; c^2 = a^2 - b^2; e = c/a; LR = 2b^2/a; 0 < e < 1 4. Hyperbola: x^2/a^2 - y^2/b^2 = 1; c^2 = a^2 + b^2; e = c/a; LR = 2b^2/a; e > 1 5. Parabola opening: y^2 = 4ax (right), y^2 = -4ax (left), x^2 = 4ay (up), x^2 = -4ay (down) 6. Ellipse foci: (+/-ae, 0) when a > b; (0, +/-be) when b > a 7. Hyperbola asymptotes: y = +/-(b/a)x 8. Rectangular hyperbola: a = b, e = sqrt(2) 9. Degenerate conics: point, line, or two intersecting lines through vertex 10. Conic definition: PF = e * PD (distance from focus = e times distance from directrix)