Conic Sections — Mathematics Class 11 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Conic Sections (Class 11) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 4 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Conic Sections (CBSE & HBSE)
Circle, parabola, ellipse, hyperbola — standard forms and key properties. CBSE expects 6–8 marks across MCQ and short/long questions on equation derivation, foci, vertices, eccentricity.
Circle
Definition
A circle is the locus of points equidistant from a fixed point (centre).
Standard Equation
Circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r².
Circle Centred at Origin
x² + y² = r².
General Equation
x² + y² + 2gx + 2fy + c = 0.
- Centre = (−g, −f).
- Radius = √(g² + f² − c).
- Real circle iff g² + f² − c > 0.
- Point circle (radius 0) iff g² + f² − c = 0.
- Imaginary circle iff g² + f² − c < 0.
Finding Equation from Conditions
| Given | Approach |
|---|---|
| Centre + radius | direct (x − h)² + (y − k)² = r² |
| Centre + point on circle | r = distance from centre to point |
| Three points | substitute into general form, solve linear system |
| Diameter endpoints (x₁,y₁), (x₂,y₂) | (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0 |
Position of a Point w.r.t. Circle
For circle S = x² + y² + 2gx + 2fy + c = 0 and point (x₁, y₁): Let S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c.
| S₁ | Point lies |
|---|---|
| < 0 | inside circle |
| = 0 | on circle |
| > 0 | outside circle |
Tangent Length from External Point
For point (x₁, y₁) external to circle: Length of tangent = √(S₁).
Common Conditions
- Circle touches x-axis: |k| = r ⇒ k = ±r (centre at (h, ±r)).
- Circle touches y-axis: |h| = r.
- Circle through origin: c = 0 in general form.
CBSE Tip: When using the general form, always compute centre and radius separately and write each clearly — examiners award marks for each.
Parabola
Definition
A parabola is the locus of points equidistant from a fixed point (focus) and a fixed line (directrix).
Standard Forms (vertex at origin)
| Equation | Opens | Focus | Directrix | Axis | Length of LR |
|---|---|---|---|---|---|
| y² = 4ax | right | (a, 0) | x = −a | x-axis | 4a |
| y² = −4ax | left | (−a, 0) | x = a | x-axis | 4a |
| x² = 4ay | up | (0, a) | y = −a | y-axis | 4a |
| x² = −4ay | down | (0, −a) | y = a | y-axis | 4a |
(LR = latus rectum: chord through focus perpendicular to axis.)
Key Terms
- Focus: fixed point.
- Directrix: fixed line. Distance from any point on parabola = distance from focus = distance from directrix.
- Vertex: midpoint of focus and directrix (here at origin).
- Axis: line through vertex and focus.
- Latus rectum: chord through focus perpendicular to axis, length 4a.
General Parabola (vertex shifted)
- (y − k)² = 4a(x − h): vertex (h, k), opens right.
- (x − h)² = 4a(y − k): vertex (h, k), opens up.
Eccentricity
For parabola, e = 1. (Eccentricity is the ratio distance from focus / distance from directrix; for parabola this is 1.)
Reflective Property (Application)
Any ray parallel to the axis striking the parabola is reflected through the focus. Used in satellite dishes, headlamps, telescopes.
Parametric Equations
For y² = 4ax: x = at², y = 2at (parameter t ∈ ℝ).
Standard Construction Steps
- Identify orientation (which direction opens).
- Compare with the right standard form to get 4a.
- Read off focus, directrix, latus rectum.
HBSE Tip: Sketch the parabola with focus, vertex, and directrix labelled — many board questions ask specifically for the diagram.
Ellipse
Definition
An ellipse is the locus of points the sum of whose distances from two fixed points (foci) is constant (= 2a).
Standard Equation (centre at origin, foci on x-axis)
x²/a² + y²/b² = 1, with a > b > 0.
- a = semi-major axis (along x-axis).
- b = semi-minor axis (along y-axis).
- c = √(a² − b²): distance from centre to focus.
- Foci: (±c, 0).
- Vertices: (±a, 0).
- Minor vertices: (0, ±b).
- Eccentricity e = c/a, 0 ≤ e < 1.
- Length of latus rectum = 2b²/a.
Foci on y-axis (Vertical Major Axis)
x²/b² + y²/a² = 1, with a > b > 0. Foci at (0, ±c), vertices at (0, ±a), minor at (±b, 0).
Sum of Distances Property
For any point P on the ellipse: PF₁ + PF₂ = 2a, where F₁ and F₂ are the foci.
Eccentricity Interpretation
- e close to 0: ellipse is nearly a circle.
- e close to 1: ellipse is highly elongated.
- e = 0 ⇒ circle (a = b).
Real-World Applications
- Planetary orbits (Kepler's first law): ellipses with the sun at one focus.
- Whispering galleries: sound from one focus is reflected to the other.
- Lithotripsy: kidney-stone breaking using elliptical reflectors.
Parametric Form
For x²/a² + y²/b² = 1: x = a cos θ, y = b sin θ (θ ∈ [0, 2π)).
Standard Solving Approach
- Identify the major axis (compare a² vs b²).
- Compute c = √(a² − b²).
- Read off foci, vertices, latus rectum, eccentricity.
CBSE Tip: Always state "foci lie on x-axis (or y-axis)" before computing — it tells the examiner you've identified orientation.
Hyperbola
Definition
A hyperbola is the locus of points the absolute difference of whose distances from two fixed points (foci) is constant (= 2a).
Standard Equation (centre at origin, foci on x-axis)
x²/a² − y²/b² = 1.
- a = semi-transverse axis.
- b = semi-conjugate axis.
- c = √(a² + b²): distance from centre to focus.
- Foci: (±c, 0).
- Vertices: (±a, 0).
- Eccentricity e = c/a, e > 1.
- Length of latus rectum = 2b²/a.
Foci on y-axis
y²/a² − x²/b² = 1. Foci at (0, ±c), vertices at (0, ±a).
Important Relation
For hyperbola: b² = a²(e² − 1) ⇒ e² = 1 + b²/a².
Asymptotes
For x²/a² − y²/b² = 1, asymptotes are y = ±(b/a) x. For y²/a² − x²/b² = 1, asymptotes are y = ±(a/b) x.
Difference Property
For any point P on the hyperbola: |PF₁ − PF₂| = 2a.
Rectangular (Equilateral) Hyperbola
When a = b: x² − y² = a². Asymptotes are y = ±x (perpendicular). Eccentricity = √2.
Conjugate Hyperbola
The conjugate of x²/a² − y²/b² = 1 is −x²/a² + y²/b² = 1, i.e. y²/b² − x²/a² = 1. It shares the same asymptotes; foci swap to the other axis.
Standard Approach
- Identify which variable has the positive sign — the transverse axis is along that variable.
- Compute c = √(a² + b²).
- Read off foci, vertices, eccentricity, latus rectum.
HBSE Tip: Distinguish hyperbola from ellipse by the sign — minus (−) in the equation indicates a hyperbola; both terms positive = ellipse.
Frequently asked questions
Are these Conic Sections notes free?
Yes — the Conic Sections notes for Mathematics (Class 11) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Conic Sections notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Conic Sections chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Conic Sections.