Q.Find the eccentricity of an ellipse if the distance between its directrix is three times the distance between its foci.

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. …