Latus Rectum

Latus Rectum of a Hyperbola

The latus rectum is a line segment that runs perpendicular to the transverse axis, passing through one of the foci, with its endpoints lying on the hyperbola. This is exactly analogous to the latus rectum you studied in the ellipse — the same geometric idea, just applied to the hyperbola now.

For a hyperbola of the standard form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the foci are at $(\pm c, 0)$ where $c^2 = a^2 + b^2$. The latus rectum through the focus $(c, 0)$ is a vertical line $x = c$. To find its endpoints, substitute $x = c$ into the hyperbola equation:

Since $c^2 = a^2 + b^2$, we get:

The endpoints of the latus rectum through $(c, 0)$ are therefore $\left(c, \frac{b^2}{a}\right)$ and $\left(c, -\frac{b^2}{a}\right)$. The length of the latus rectum is the distance between these two points:

The same length applies for the latus rectum through the other focus $(-c, 0)$ as well, by symmetry.

For a hyperbola of the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (vertical transverse axis), the latus rectum runs horizontally through the foci $(0, \pm c)$, and its length is again $\frac{2b^2}{a}$. …