MathematicsCoordinate geometryClass 11Class 12

Conic sections & eccentricity: interactive simulation

A conic section is the locus of a point P whose distance from a fixed point (focus S) is always a constant ratio e times its perpendicular distance from a fixed line (directrix): SP = e · PM. That single number e (the eccentricity) determines the curve’s entire shape and nature.

Useful for: JEE Main, JEE Advanced, CBSE Boards

Try this

Vary e with the slider or press Animate to watch the curve morph continuously: a circle (e = 0) flattens into an ellipse (0 < e < 1), bursts open into a parabola (e = 1), and widens into a two-branched hyperbola (e > 1). Drag point P to see SP = e · PM verified at every spot, or switch to 3D Cone Slice to see the cutting plane angle β.

  1. 1

    Set e = 0 and observe the shape and directrix.

    What you should see

    You get a perfect circle. The directrix is at infinity, both foci coincide at the centre, and SP is constant (the radius).

  2. 2

    Slide e slowly from 0.0 to 0.95.

    What you should see

    The circle stretches into an ellipse. The second focus and directrix move in from infinity as the ellipse elongates.

  3. 3

    Hit e = 1.00 (or click Parabola preset).

    What you should see

    The ellipse opens at infinity and becomes a single open parabola. Distance from vertex to focus equals distance from vertex to directrix.

  4. 4

    Increase e beyond 1 (e.g. e = 1.414).

    What you should see

    Two separate hyperbola branches appear. Dotted asymptotes emerge through the centre with slopes ±√(e² − 1).

  5. 5

    Drag point P around the curve in Focus-Directrix mode.

    What you should see

    Even though SP and PM change continuously, the ratio SP / PM always stays exactly equal to e.

  6. 6

    Switch to the 3D Cone Slice view and tilt the plane angle β.

    What you should see

    When β = 90° it cuts a circle; β > α gives an ellipse; β = α cuts a parabola; β < α cuts both nappes forming a hyperbola.

Formulas behind it

Focus-Directrix definition

SPPM=e  ⟹  SP=e⋅PM\frac{SP}{PM} = e \implies SP = e \cdot PM

Polar equation (focus at origin)

r(θ)=l1−ecos⁡θr(\theta) = \frac{l}{1 - e\cos\theta}

Ellipse (0 < e < 1)

x2a2+y2b2=1,b2=a2(1−e2)\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad b^2 = a^2(1 - e^2)

Parabola (e = 1)

y2=4ax,e=1y^2 = 4ax, \quad e = 1

Hyperbola (e > 1)

x2a2−y2b2=1,b2=a2(e2−1)\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \quad b^2 = a^2(e^2 - 1)

Cone cutting plane angle

e=cos⁡βcos⁡αe = \frac{\cos\beta}{\cos\alpha}

Common mistakes

  • Thinking a parabola has an eccentricity of 0. A circle has e = 0; a parabola has e = 1.
  • Confusing the eccentricity formula: for an ellipse b² = a²(1 − e²), while for a hyperbola b² = a²(e² − 1).
  • Assuming the directrix passes through the vertex. The vertex is midway between the focus and the directrix for a parabola (SP = PM).
  • Forgetting that a hyperbola has two branches — the locus SP = e·PM with distance taken absolute generates both branches with the two directrices.
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FAQ

Why is the eccentricity of a circle 0?

In an ellipse b² = a²(1 − e²). For a circle, the semi-major and semi-minor axes are equal (a = b), so 1 − e² = 1, meaning e = 0. The directrix is infinitely far away.

What happens at e = 1?

The conic transitions from a closed curve (ellipse) to an open curve (parabola). The second vertex and focus move off to infinity.

What is a rectangular hyperbola?

A hyperbola whose asymptotes are perpendicular (at 90° to each other). Since slope m = ±b/a, b = a, giving e = √(1 + b²/a²) = √2 ≈ 1.414.

How does eccentricity relate to a cone slice?

If a cone has semi-vertical angle α and a cutting plane is inclined at angle β to the cone axis, the cross-section has eccentricity e = cos β / cos α.

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