5. The Ellipse

Why study ellipses?

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

earth

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

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Horizontal major axis
Vertical major axis
Centre other than origin

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 Flash animation.

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Ellipses with Horizontal Major Axis

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

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

Our complete ellipse is formed:

pencil ellipse

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

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

.

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

math expression

Let's see this in LiveMath.

LIVEMath

Example 1 - Ellipse with Horizontal Major Axis

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Find the coordinates of the vertices and foci of

Sketch the curve.

Answer

Ellipse with Vertical Major Axis

math expression

Our first example above had a horizontal major axis.

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

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

Example 2.

Find the coordinates of the vertices and foci of

25x² + y² = 25.

Sketch the curve.

Answer

Example 3.

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

Answer

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?

Answer

Ellipses with Centre Other Than the Origin

Like the other conics, we can move the ellipse so the 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: