Conic sections and eccentricity

Eccentricity

How can one number turn a circle into an ellipse, a parabola and a hyperbola?

Every conic is the set of points whose distance to a focus is e times their distance to a directrix. Drag e from 0 to 2 and watch the curve change family at e = 1.

Drag the ringed points on the board.

High school
University
Focus and directrix
Eccentricity
Ellipse, parabola, hyperbola
Polar equation

e

ellipse

parabola

hyperbola

0.60

\theta_P

2.00

|PF| = e \cdot |PD|

$e$: 0.60
$|PF|$: 1.600
$|PD|$: 2.667
$|PF| / |PD|$: 0.600

Closed curveyes: an ellipse

Guided tour

1e = 0: a circle
With $e=0$ the directrix is infinitely far away and every point is the same distance, $2$ , from the focus $F$ . The curve is a circle.
20 < e < 1: an ellipse
At $e=0.5$ every point is half as far from $F$ as from the green directrix. Check the readout: $|PF|/|PD|=0.5$ wherever $P$ is. The circle has stretched into a closed ellipse.
3e = 1: a parabola
At $e=1$ the two distances are equal. The far end of the ellipse has run off to infinity and the curve opens up: $y^2=4(x+1)$ .
4e > 1: a hyperbola
At $e=1.5$ points are farther from $F$ than from the directrix, and the curve splits into two branches. Move $\theta_P$ and the ratio stays $1.5$ on both of them.

Explanation

1
One definition for all three
Fix a point $F$ (the focus), a line $D$ (the directrix) and a number $e\ge0$ (the eccentricity). The conic is every point $P$ with
$|PF| = e\cdot|PD|$
- $e<1$ : ellipse (the circle is the limit $e=0$ )
- $e=1$ : parabola
- $e>1$ : hyperbola
2
The polar equation behind the board
Put the focus at the origin and the directrix at $x=-\frac{\ell}{e}$ . The definition becomes
$r=\frac{\ell}{1-e\cos\theta}$
where $\ell$ is the semi-latus rectum (here $\ell=2$ ). When $e\ge1$ the denominator can reach $0$ : that is the direction in which the curve escapes to infinity.
3
Eccentricity in the standard equations
For the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with $a>b$ , the foci are at $(\pm c,0)$ with $c^2=a^2-b^2$ and $e=\frac{c}{a}<1$ . For the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ , $c^2=a^2+b^2$ and $e=\frac{c}{a}>1$ . For the parabola $y^2=2px$ , $e=1$ .
The closer $e$ is to $0$ , the rounder the ellipse; the closer to $1$ , the flatter.
4
Why they are called conic sections
Slice a double cone with a plane. A cut across one nappe gives an ellipse, a cut parallel to a side line gives a parabola, and a steep cut through both nappes gives a hyperbola. The tilt of the plane against the cone sets $e$ , which is why the same three shapes come out of both definitions.
The focus also explains their optics: a parabolic dish sends incoming parallel rays to its focus, and light from one focus of an ellipse reflects to the other.

Check yourself

Q01

What is the eccentricity of $\frac{x^2}{25}+\frac{y^2}{16}=1$ ?

Q02

On a parabola, how does $|PF|$ compare with the distance from $P$ to the directrix?

Q03

A conic has $e=\sqrt2$ . Which curve is it?

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