Orbiting satellites (including the earth revolving around the sun, and the moon revolving around us) trace out elliptical paths.
Many buildings and bridges use the ellipse as a pleasing (and strong) shape.
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 what this means in the following animation.
Ellipse with horizontal major axis (in blue) and showing the minor axis in magenta color.
where `a` is the length from the center of the ellipse to the end the major axis, and `b` is the length from the center to the end of the minor axis.
The vertices of an ellipse are at (−a, 0) and (a, 0).
Ellipse showing vertices and foci.
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.
Here, `a^2= 100`, so `a = ± 10`, so the vertices are at `(-10, 0)` and `(10, 0)`.
So the foci are at `(-6, 0)` and `(6, 0)`.
Ellipse with vertical major axis (in blue) and minor axis in magenta.
A vertical major axis means the ellipse will have greater height than width.
We always choose our a and b such that a > b. The major axis is always associated with a.
So `b = 1` and `a = 5`. In this example, the major axis is vertical.
So the vertices are at `(0, -5)` and `(0, 5)`.
So the foci are at `(0, -4.9)` and `(0, 4.9)`.
The foci (green dots) are very close to the vertices in this ellipse.
Find the equation of the ellipse which has a minor axis of length 8 and a vertex at (0,-5).
Ellipse with minor axis of length 8 and vertex at (0,-5).
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.
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 closest we are to the sun is when we are at the vertex closest to the sun (point (150, 0) in the diagram below), and the furthest we are is when we are at the other vertex (point (−150, 0)).
Note: In the graph, I have exaggerated the shape of the elliptical orbit (I squashed it a bit so it looks more elliptical) and the positions for the 2 foci are shown much further apart than they really are.
In the diagram above, units are in millions of kilometers.
The distance a = OA = 149,597,871 is the length of the semi-major axis of our ellipse.
We have assumed that the sun is at the right-hand focus, at the point (c, 0).
The vertices are at A (149 597 871, 0) and B (−149 597 871, 0).
We need to find c, to tell us where the foci are.
`= 147\ 104\ 573\ "km"
The foci are `2 × c = 2 × 2\ 493\ 298` ` = 4\ 986\ 596` km apart.
The radius of the sun is around `1\ 400\ 000` km, so the 2nd focus is not all that much further out than the surface of the sun.
Our orbit is almost circular. (The eccentricity is very small at `1/60`).
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.
Ellipse with center (h, k).
We first observe that the centre of the ellipse will be at `(1, -2)`. The major axis will have length `10` (since `a = 5`) and the minor axis will be length `6` (since b = 3).
How can we obtain an ellipse from slicing a cone?