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Class 11 Maths Notes

Conic Sections Class 11 Notes

Complete, exam-ready notes on conic sections: how slicing a cone produces circles, parabolas, ellipses and hyperbolas, the standard equations for each, foci, eccentricity that classifies them, and latus rectum lengths — written for CBSE boards and JEE revision.

Class11SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

Conic sections are the four curves — circle, parabola, ellipse, hyperbola — obtained by slicing a double cone at different angles, each defined by eccentricity.

Circles

Circle

The set of all points at a fixed distance r from a centre (h, k). The standard equation is: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2. When centred at the origin (h = k = 0), it simplifies to x2+y2=r2x^2 + y^2 = r^2.

  • General form: x² + y² + 2gx + 2fy + c = 0, with centre (−g, −f) and radius √(g² + f² − c).
  • A real circle exists only when g² + f² − c > 0.
  • Eccentricity of a circle is e = 0 — all points are equidistant from the centre.

Parabola

Parabola

The set of all points equidistant from a fixed point (focus) and a fixed line (directrix). For the standard form y2=4axy^2 = 4ax, the focus is at (a, 0), the directrix is x = −a, and the latus rectum has length 4a.

  • y² = 4ax opens right; y² = −4ax opens left.
  • x² = 4ay opens upward; x² = −4ay opens downward.
  • Eccentricity e = 1 for every parabola.
  • The latus rectum passes through the focus, perpendicular to the axis.

Ellipse

Ellipse

The set of all points whose sum of distances from two foci is constant. The standard form (a > b) is: x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. The major axis has length 2a, the minor axis 2b.

  • Foci at (±ae, 0) where eccentricity e = √(1 − b²/a²), so 0 < e < 1.
  • Vertices at (±a, 0); minor-axis vertices at (0, ±b).
  • Latus rectum length = 2b²/a.
  • The relation b² = a²(1 − e²) connects a, b and e.

Quick check

If the denominator under x² is larger, the major axis is horizontal. If the denominator under y² is larger, it is vertical — just compare a² and b².

Hyperbola

Hyperbola

The set of all points whose difference of distances from two foci is constant. The standard form is: x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. The vertices are at (±a, 0) and the foci at (±ae, 0).

  • Eccentricity e = √(1 + b²/a²), so e > 1 always.
  • Latus rectum length = 2b²/a.
  • Asymptotes: y = ±(b/a)x — the curve approaches but never touches these lines.
  • Conjugate hyperbola: x²/a² − y²/b² = −1 (swap the sign).

Identifying Conics by Eccentricity

  • e = 0: circle.
  • e = 1: parabola.
  • 0 < e < 1: ellipse.
  • e > 1: hyperbola.
  • This single parameter e classifies all four conics — a key JEE shortcut.

JEE trap

A circle is technically an ellipse with e = 0 and both foci coinciding at the centre. JEE questions occasionally exploit this by asking 'which eccentricity gives a circle?'

Parametric Equations of Conics

Parametric form

A parametric equation represents each coordinate as a function of a parameter t (or θ), making it easy to plot points or compute tangents. For the standard parabola y2=4axy^2 = 4ax, the parametric form is x=at2,  y=2atx = at^2,\; y = 2at.

  • Ellipse: x = a cos θ, y = b sin θ (θ is the eccentric angle, 0 ≤ θ < 2π).
  • Hyperbola: x = a sec θ, y = b tan θ (θ is the eccentric angle).
  • Parametric forms are especially useful in JEE for finding tangent lines and normals.
  • Substituting the parameter into the standard equation verifies the parametric pair always lies on the curve.

Solved Examples

Example: Find the eccentricity, foci and latus rectum of the ellipse x225+y216=1\frac{x^2}{25} + \frac{y^2}{16} = 1.

Solution: Here a² = 25 (a = 5), b² = 16 (b = 4). Eccentricity e = √(1 − 16/25) = √(9/25) = 3/5. Foci at (±5 × 3/5, 0) = (±3, 0). Latus rectum = 2b²/a = 2(16)/5 = 32/5 units.

Example: A parabola has focus (2, 0) and directrix x = −2. Find its equation.

Solution: The focus is at (a, 0) with a = 2, so the standard form is y² = 4ax. Substituting: y² = 8x.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Circle (centred at origin)

x2+y2=r2x^2 + y^2 = r^2

Circle (general centre)

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

Parabola

y2=4axy^2 = 4ax

Ellipse

x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Ellipse eccentricity

e=1−b2a2e = \sqrt{1 - \frac{b^2}{a^2}}

Hyperbola

x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Hyperbola eccentricity

e=1+b2a2e = \sqrt{1 + \frac{b^2}{a^2}}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • Circle has e = 0, parabola e = 1, ellipse 0 < e < 1, hyperbola e > 1.
  • Parabola latus rectum = 4a; always the perpendicular chord through the focus.
  • Ellipse: b² = a²(1 − e²); this relation connects a, b and e.
  • Hyperbola eccentricity is always > 1; never forget the + sign under the root.
  • Identify the conic first from the equation before applying formulas — saves time.
  • For latus rectum of ellipse and hyperbola, both use 2b²/a — but eccentricity formulas differ.
  • General second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0: check discriminant B² − 4AC to classify.
  • JEE loves shifted conics — complete the square to bring any conic to standard form.

FAQ

Common questions

What is eccentricity?

Eccentricity e is a number that classifies a conic: e = 0 is a circle, e = 1 is a parabola, 0 < e < 1 is an ellipse, and e > 1 is a hyperbola.

What is the latus rectum?

The latus rectum is a chord through the focus, perpendicular to the axis of symmetry. For a parabola it is 4a; for an ellipse and hyperbola it is 2b²/a.

How do you find the focus of a parabola from its equation?

For y² = 4ax, the focus is at (a, 0) and directrix is x = −a. Identify a by comparing your equation with this standard form.

What is the difference between an ellipse and a hyperbola?

An ellipse has 0 < e < 1 and a + b sum of focal distances; a hyperbola has e > 1 and a constant difference of focal distances. Their equations differ by a sign.

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