An ellipse is a conic section, that resembles an oval, but is formally characterized by the following property: there exist two points and inside the ellipse (called focal points) such that for every point on the ellipse, the quantity is constant where denotes the distance from to Ellipses are essentially circles that have been stretched, generally (not necessarily) along one axis. They are important objects in coordinate geometry, Euclidean geometry and number theory.
Suppose two points and are given, and one wishes to determine the locus of points such that is some constant . The points and are called the foci of the ellipse (singular: focus). To simplify the calculations, one assumes and . Given the equation when and are of this form, one may retrieve the more general equation by rotating, dilating, and translating accordingly.
The condition imposed is precisely
Isolating the leftmost radical and squaring both sides gives
Again, isolating the remaining radical and simplifying yields
Finally, squaring both sides and rearranging gives
Setting , the equation is simply
The equation of an ellipse centered at the origin is
where and are real numbers. More generally, an ellipse centered at will have an equation of the form
The axis of the ellipse through and is called the major axis of the ellipse, and the axis perpendicular to the major axis is the minor axis. In the above notation, the length of the major axis is , since the ellipse meets the -axis at precisely and . An analogous observation shows the length of the minor axis is . The parameter is called the focal distance of the ellipse and represents the distance from a focus to the center of the ellipse.
The eccentricity of the ellipse is defined as . This can be thought of as measuring how much the ellipse deviates from being a circle; the ellipse is a circle precisely when , and otherwise one has .
A tunnel opening is shaped like a half ellipse. The tunnel is 80 units wide and 25 units tall. Find the equation of the ellipse assuming it is centered at the origin.
The length of the major axis (which is on the -axis since it's width) is 40 units, and the length of the minor axis (which is on the -axis since it's height) is 25 units.
Since the ellipse (or half ellipse) is centered at the origin, the equation is
Intuitively, an ellipse with major axis of length and minor axis of length is simply a circle of radius that has been squished/stretched along the -axis by a factor of . Accordingly, the area enclosed by this ellipse should be .
Although this is not a rigorous proof, the intuition is not difficult to turn into a precise argument. The upper half of this ellipse has equation
so the area of the ellipse is
But is just half the area of the circle with equation , which equals . Thus,
If an ellipse's area is the same as the area of a circle with radius 4, what is the product of the ellipse's major and minor axes?
First, we would like to find the area of the circle with radius 4. Using the area formula of a circle, we get
Now that we know the area formula of an ellipse is , we get that . Note that and are the major and minor radii of the ellipse, and what we actually want are the major and minor axes, which are and .
Our last step is to find the product of and
In astronomy, Kepler's laws state that the orbit of a planet around the sun traces an ellipse, one of whose foci is the sun itself. Furthermore, information about this ellipse can quantify the orbital period of the planet (how much time it takes for the planet to go once around the sun).
If is the orbital period and the ellipse corresponding to this orbit has a major axis of length , then , where indicates direct proportionality.
The planet Xabros is orbiting the sun in an elliptical path with a major axis of length . Doofenschmirz has a machine that can scale the length of the ellipse's major axis by a factor of : that is, after using the machine, Xabros will be in a path with a major axis of length .
If Doofenschmirz's evil plan is to double the orbital period of Xabros, what is