Example - determining the elements

Let us see on a concrete case how all the elements are read off starting from the canonical equation.

Example — Elements of an ellipse

Given $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$, determine the semi-axes, foci and eccentricity.

$a^2 = 25,\ b^2 = 9$: hence $a = 5$, $b = 3$ (semi-major axis along $x$).

$c^2 = a^2 - b^2 = 16 \implies c = 4$. Foci $F_{1,2}(\pm 4; 0)$.

$e = c/a = 4/5 = 0{,}8$. Directrices $x = \pm a^2/c = \pm 25/4$.

Links

Topics: Ellipse
Concepts: Directrices · Eccentricity · Ellipse · Foci · Semi-axes
Skills: Calculating · Analytic geometry