Ellipse — Foci, Eccentricity & Focal Distances

For $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ with $a > b$ (major axis along $x$):

• Foci: $(\pm c, 0)$ with $c^2 = a^2 - b^2$; distance between foci $= 2c$.
• Eccentricity: $e = \dfrac{c}{a} = \sqrt{1 - \dfrac{b^2}{a^2}}$ ($0 < e < 1$).
• Major axis is along the variable with the LARGER denominator. If $b > a$, swap roles: the major axis is along $y$, foci at $(0, \pm c)$.
• Parametric point: $(a\cos\theta,\ b\sin\theta)$.

For any point $P$ on the ellipse with foci $F_1, F_2$:

• This is the locus definition: 'sum of distances from two fixed points is constant'.
• The latus rectum has length $\dfrac{2b^2}{a}$, with endpoints at $\left(\pm c, \pm \tfrac{b^2}{a}\right)$.

Translate the given data into equations in $a, b, c$ (using $c = ae$, $c^2 = a^2 - b^2$, latus rectum $= 2b^2/a$):

• Vertices and foci given: read $a$ from the vertices, $c$ from the foci, then $b^2 = a^2 - c^2$.
• Two points given: substitute into $\tfrac{x^2}{A} + \tfrac{y^2}{B} = 1$ and solve the linear system in $\tfrac1A, \tfrac1B$.
• Position of a point $(x_1,y_1)$: inside if $\tfrac{x_1^2}{a^2}+\tfrac{y_1^2}{b^2} < 1$, on if $=1$, outside if $>1$.
• Area enclosed $= \pi a b$.