Q.Find the

Several powerful derived facts follow from the ellipse's standard equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ($b^2=a^2(1-e^2)$, $a>b$):

• The latus rectum — the focal chord perpendicular to the major axis — has length $\dfrac{2b^2}{a}$, with end points $\left(ae,\pm\dfrac{b^2}{a}\right)$.
• The distance between the two directrices is $\dfrac{2a}{e}$.
• Most famously, the sum of the focal distances of any point on the ellipse is the constant $2a$ — exactly the length of the major axis. This is the mathematical basis of the classical "pin and string" construction: loop a string of length $2a$ around two pins placed at the foci, and a pencil kept taut against the string traces out the ellipse.
• Points on the ellipse can be described using the eccentric angle $\theta$ via the parametric form $x=a\cos\theta,\,y=b\sin\theta$, obtained by relating each point on the ellipse to the corresponding point on its auxiliary circle $x^2+y^2=a^2$ (the circle drawn on the major axis as diameter) sharing the same $\theta$. Importantly, $\theta$ is generally not the actual angle $\angle POX$ that the radius to the point makes with the axis — it is a separate, purely parametric angle.

These identities — especially the constant-sum property and the eccentric-angle parametrisation — are the standard toolkit for solving "find the eccentricity/foci/latus rectum given..." problems and for setting up locus problems involving points at given eccentric angles.