Q.Find the equation of the ellipse in standard form if eccentricity is 2/3 and passes through $(2, -5/3)$.

Applying the same focus-directrix idea as the parabola, but now with eccentricity $0<e<1$, and choosing the midpoint of the two points that internally/externally divide the focus-to-directrix-foot segment in the ratio $e:1$ as the origin, the algebra reduces to the standard equation of an ellipse:

Unlike the parabola, the ellipse has two foci, $(\pm ae,0)$, and two directrices, $x=\pm a/e$ — a direct consequence of the curve being symmetric not just about one axis but about both the major axis (length $2a$, the longer one) and the minor axis (length $2b$). When $b>a$ instead, the roles of the axes swap: the major axis runs along $Y$ instead of $X$, and every formula mirrors acc …