Elements of 9x²+4y²=36

Problem

Given $9x^2 + 4y^2 = 36$, determine the semi-axes, foci and eccentricity (careful: which semi-axis is the major one?).

Solution

We bring the equation into canonical form by dividing by $36$: $\frac{9x^2}{36} + \frac{4y^2}{36} = 1 \iff \frac{x^2}{4} + \frac{y^2}{9} = 1.$

Here the denominators are $4$ (under $x^2$) and $9$ (under $y^2$): the larger one is under $y^2$, hence the semi-major axis is along the $y$-axis. Consequently: $\text{semiasse minore (lungo } x) = \sqrt{4} = 2, \qquad \text{semiasse maggiore (lungo } y) = \sqrt{9} = 3.$

The foci lie on the major axis, that is on the $y$-axis, with $c = \sqrt{9 - 4} = \sqrt{5} \implies F_{1,2}(0;\ \pm\sqrt{5}).$

The eccentricity is computed by dividing $c$ by the semi-major axis: $\boxed{e = \frac{\sqrt{5}}{3} \approx 0{,}75}.$

Links

Topics: Ellipse
Concepts: Eccentricity · Ellipse · Foci · Semi-axes
Skills: Calculating · Analytic geometry
Exercise type: Calculating expressions