Ellipse - Equation, Formula, Properties, Graphing

Book A Free Math Class

Ellipse - Equation, Formula, Properties, Graphing

TL;DR

An ellipse is the set of all points in a plane whose distances to two fixed points (called foci) sum to a constant. Its standard equation is ( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 ) where ( a ) is the semi-major axis and ( b ) is the semi-minor axis.

What Is an Ellipse?

An ellipse is the locus of all points in a plane whose distances to two fixed points — the foci — sum to a constant value. Mathematically, for any point ( P ) on the ellipse:

[ |PF_1| + |PF_2| = 2a ]

where ( F_1 ) and ( F_2 ) are the foci and ( 2a ) is the length of the major axis.

An ellipse is also a conic section — formed by slicing a cone with a plane at a slight angle. A circle is the special case of an ellipse where both foci coincide at the center. The standard equation of an ellipse centered at the origin with major axis along the x-axis is:

[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b > 0 ]

where ( a ) is the semi-major axis length and ( b ) is the semi-minor axis length.

What Are the Parts of an Ellipse?

Part Definition
Centre The midpoint of the major axis (and minor axis)
Foci (( F_1, F_2 )) Two fixed points whose distance-sum defines the curve
Major axis Longest diameter, length ( 2a ), contains both foci
Minor axis Shortest diameter perpendicular to major axis, length ( 2b )
Vertex Endpoint of the major axis
Co-vertex Endpoint of the minor axis
Eccentricity (( e )) Measure of how "stretched" the ellipse is, ( 0 \le e < 1 )

What Is the Equation of an Ellipse?

Standard Form — Centered at Origin

For an ellipse centered at ( (0,0) ) with major axis along the x-axis: [ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b ] For major axis along the y-axis: [ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1, \quad a > b ]

Shifted Form — Centered at ( (h,k) )

[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 ]

Relationship Between ( a, b, ) and ( c )

The distance from the center to each focus is ( c ), where: [ c = \sqrt{a^2 - b^2} ] So the foci sit at ( (\pm c,0) ) for a horizontal ellipse, or ( (0,\pm c) ) for a vertical ellipse.

How Do You Derive the Ellipse Equation?

The standard form ( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 ) falls out of the focus-pair definition + the distance formula. Here's the derivation.

Setup. Place the center at the origin, foci at ( F_1=(-c,0) ) and ( F_2=(c,0) ) and let ( P=(x,y) ) be any point on the ellipse. By definition, the sum of distances to the foci is ( 2a ): [ \sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2a ]

Isolate one radical. [ \sqrt{(x + c)^2 + y^2} = 2a - \sqrt{(x - c)^2 + y^2} ]

Square both sides. [ (x+c)^2+y^2=4a^2−4a(x−c)^2+y^2+(x−c)^2+y^2 ]

Solve for remaining radical. [ a(x−c)^2+y^2=a^2−cx ]

Square again. [ a^2[(x−c)^2+y^2]=a^4−2a^2cx+c^2x^2 ]

Expand and group. [ (a^2−c^2)x^2+a^2y^2=a^2(a^2−c^2) ]

Substitute ( b^2=a^2−c^2 ): [ b^2x^2+a^2y^2=a^2b^2 ]

Divide by ( a^2b^2 ): [ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 ]

What Are the Properties of an Ellipse?

  1. Two axes of symmetry. The major and minor axes are both lines of symmetry.
  2. Two foci, constant distance-sum. For every point ( P ) on the curve, ( |PF_1| + |PF_2| = 2a ).
  3. Closed, bounded curve. An ellipse is a finite closed curve, enclosing a finite area.
  4. Area = ( \pi ab ). Generalizes the circle's ( \pi r^2 ).
  5. Perimeter — no closed-form formula. Ramanujan's approximation gives ( P \approx \pi \left[ 3(a + b) - \sqrt{(3a + b)(a + 3b)} \right] ), accurate to a few parts per million for moderate eccentricities.
  6. Reflective property. Any ray emitted from one focus reflects off the ellipse and passes through the other focus.

How Do You Draw an Ellipse? (The Gardener's Method)

Materials: Two pins, a piece of string of length ( 2a ), and a pencil.

Steps:

  1. Decide the major-axis length ( 2a ) and the focal distance ( 2c ).
  2. Press the two pins into the drawing surface, ( 2c ) apart.
  3. Tie the string into a loop of total length ( 2a + 2c ).
  4. Pull the string taut with the pencil inside the loop and trace a complete circuit.

What Is Eccentricity?

The eccentricity ( e ) of an ellipse measures how elongated it is.
[ e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}} ]

How Do You Graph an Ellipse?

For an ellipse ( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 ):

Why Are Ellipses Important?

The ellipse was one of the most important geometric discoveries in the history of science.

Worked Example of Ellipse

Find the foci of the ellipse ( \frac{x^2}{25} + \frac{y^2}{16} = 1 ).
Correct method:
[ c = \sqrt{25 - 16} = 3 ]
The foci are at ( (\pm 3,0) ).
Eccentricity:
[ e = \frac{3}{5} = 0.6 ]

The Mathematicians Who Shaped Ellipse Theory

Frequently Asked Questions