A Nice Problem about the Focal Chord of an Ellipse
Focal chord of ellipse is a chord that passes through focus. So there is a relationship between the length of the focal chord and the length of another parallel chord passing through the center of the ellipse.
Consider that in Figure 1 the closed line is an ellipse and the lines between points M and M’ and D and D’ are segments which pass through the focal point F and through the center of the ellipse C. The problem is to prove the relation:
where 2a corresponds to the length of the major axis of the ellipse.
Here our solution
As a resolution strategy we will use polar coordinates.
We will adopt the following convention:
- Semi-major axis: a;
- Semi-minor axis: b;
- Distance from center C to focus F: f.
For the coordinates the point D:
and
Analogously,
and
Applying the coordinates of point D in the reduced equation of the ellipse, we have:
and using polar coordinates
Thus
Using the Pythagorean theorem, we have:
thus developing step by step
and let’s split this fractional expression into two partial fractions,
but, in M, we have the possibility to do that
and
or
combining this with the quadratic expression of MF, we have
We can also make the following articulation
or, factoring and simplifying
Analogously, taking into account the symmetry properties of the ellipse, we can do
We can use these two results in the expression of partial fractions obtained earlier, and doing that we will have
the sum inside the parentheses can be written as
what makes us write:
