Find the curves in which the coordinate planes cut the…

Find the curves in which the coordinate planes cut the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 $ and sketch them.

Explanation of Method

The sections of an ellipsoid by the three coordinate planes.

Step 1: Setting one coordinate to zero

plane	section	semi-axes
$z=0$	$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$	$a,\ b$
$y=0$	$\dfrac{x^2}{a^2}+\dfrac{z^2}{c^2}=1$	$a,\ c$
$x=0$	$\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$	$b,\ c$

\[\text{three ellipses (the principal sections)}\]

Each ellipse lies in its own coordinate plane. Any two of them share two vertices of the ellipsoid: $(\pm a,0,0)$, $(0,\pm b,0)$ or $(0,0,\pm c)$.

If two semi-axes are equal, the section through them is a circle. For example, $a=b$ makes the $z=0$ section a circle, and the ellipsoid is then a surface of revolution about $Oz$.

$a$ $b$ $c$

$z=0:\ \tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1$
$y=0:\ \tfrac{x^2}{a^2}+\tfrac{z^2}{c^2}=1$
$x=0:\ \tfrac{y^2}{b^2}+\tfrac{z^2}{c^2}=1$

The three principal sections of the ellipsoid (drawn with $a>b>c$). Dashed parts are on the hidden side. Each pair of ellipses shares two vertices.

Result

\[\boxed{\,\text{three ellipses (the principal sections)}\,}\]