5. The Ellipse

Why study ellipses?

Orbiting satellites (including the earth and the moon) trace out elliptical paths.

earth revolving around sun

Many buildings and bridges use the ellipse as a pleasing (and strong) shape.

elliptical bridge

One property of ellipses is that a sound (or any radiation) beginning in one focus of the ellipse will be reflected so it can be heard clearly at the other focus. You can see this working in the following animation.

Ellipses with Horizontal Major Axis

ellipse with major and minor axes

The equation for an ellipse with a horizontal major axis is given by:

`x^2/a^2+y^2/b^2=1`

The ellipse is defined as the locus of a point `(x,y)` which moves so that the sum of its distances from two fixed points (called foci, or focuses) is constant.

We can produce an ellipse by pinning the ends of a piece of string and keeping a pencil tightly within the boundary of the string, as follows.

We start with these 2 foci:

pencil ellipse

We pin the ends of the string to the foci and begin to draw, holding the string tight:

pencil ellipse

pencil ellipse

pencil ellipse

pencil ellipse

Our complete ellipse is formed:

pencil ellipse

The foci (plural of 'focus') of the ellipse (with horizontal major axis)

`x^2/a^2+y^2/b^2=1`

are at (-c,0) and (c,0), where c is given by:

`c=sqrt(a^2-b^2`

The vertices of an ellipse are at (-a, 0) and (a, 0).

ellipse with vertices and foci

Example 1 - Ellipse with Horizontal Major Axis

Need Graph Paper?

rectangular grid
Download graph paper

Find the coordinates of the vertices and foci of

`x^2/100+y^2/64=1`

Sketch the curve.

Ellipse with Vertical Major Axis

ellipse with vertical major axis

Our first example above had a horizontal major axis.

If the major axis is vertical, then the formula becomes:

`x^2/b^2+y^2/a^2=1`

We always choose our a and b such that a > b. The major axis is always associated with a.

Example 2 - Ellipse with Vertical Major Axis

Find the coordinates of the vertices and foci of

`25x^2+y^2=25`

Sketch the curve.

Example 3

Find the equation of the ellipse which has a minor axis of length 8 and a vertex at (0,-5).

Eccentricity

The eccentricity of an ellipse is a measure of how elongated it is. If the eccentricity approaches value 0, the curve becomes more circular, and if it approaches 1, the ellipse becomes more elongated.

We can calculate the eccentricity using the formula:

`text(eccentricity)=c/a`

Real Example

sun
The Sun

The Earth revolves around the sun in an elliptical orbit, where the sun is at one of the foci. (This was discovered by Keppler in 1610).

The semi-major axis is approximately 149,597,871 km long and it is known that the ratio `c/a` is equal to `1/60`.

(i) What are the greatest and least distances the Earth is from the sun?

(ii) How far from the sun is the other focus?

(The "semi-major axis" means half of the major axis length. In our example, it is (close to) the "average" distance of the sun from the earth, and is also known as one A.U., or "astronomical unit".)

Ellipses with Centre Other Than the Origin

Like the other conics, we can move the ellipse so that its axes are not on the x-axis and y-axis. We do this for convenience when solving certain problems.

For the horizontal major axis case, if we move the intersection of the major and minor axes to the point (h, k), we have:

`((x-h)^2)/a^2+((y-k)^2)/b^2=1`

The ellipse is as follows:

ellipse with center (h,k)

Example 4

Sketch the ellipse with equation

`((x-1)^2)/25+((y+2)^2)/9=1`

Conic section: Ellipse

How can we obtain an ellipse from slicing a cone?

We start with a double cone (2 right circular cones placed apex to apex):

double cone

When we slice one of the cones at an angle to the sides of the cone, we get an ellipse, as seen in the view from the top (at right).

conic section -  ellipseconic section - ellipse

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