Ellipse in standard position
Theorem 4.1 (Ellipse in standard position). Let \(0 \leq c < a\) and \(b^2 = a^2 - c^2\). The set of points whose distances to the foci \(F_1(-c, 0)\) and \(F_2(c, 0)\) add up to \(2a\) is the ellipse
\[ \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1. \]
It meets the \(x\)-axis at \((\pm a, 0)\) and the \(y\)-axis at \((0, \pm b)\).