If the ellipse is placed in the Cartesian plane with the defined Cartesian coordinate system so that the centre is located at the origin , and the foci and lie on the negative and positive portions of the coordinate axis respectively, then an analytic equation of the ellipse can be derived.
Theorem 1. An equation of the ellipse with foci at and is
(1) |
where is the semimajor axis, is the semiminor axis, and .
Figure 1: Ellipse in the Cartesian plane |
Proof. Let be any point of the ellipse. Then the equation holds
from which follows
Squaring the above equation we have
Squaring the last equation we obtain
Since , then , and the equation above can be rewriten as
Dividing both sides of the above eqaution by , we receive
which is the presented equation of the ellipse.
Conversely, it can be shown that if the equation holds, then the point is located on the ellipse with the foci at and
Remark. If and are positive constants and , the Cartesian equation (1.1) is called the standard form for the equation of an ellipse with the centre at the origin and with the horizontal major axis. The standard equation of the ellipse with the same centre but with the vertical major axis is
(2) |
Figure 2: Ellipse with vertical major axis |
Theorem 2. Let the ellipse with the centre at the origin, semimajor axis a, semiminor axis b, and axes in the coordinate axes x and y be shifted so that the centre is at the point . The equation of the ellipse shifted into a new position will have one of the following standard forms
(3) |
(4) |
Figure 3: Ellipses with shifted centres |