Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
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)