Q.Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbola $9y^2 - 4x^2 = 36$.

Concept: Hyperbola Standard Form

First, divide the equation by 36 to get standard form:

This is a vertical hyperbola $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ with $a^2 = 4$ and $b^2 = 9$, so $a = 2$ and $b = 3$.

For a vertical hyperbola, $c^2 = a^2 + b^2 = 4 + 9 = 13$, giving $c = \sqrt{13}$.

The vertices lie on the transverse axis (the $y$-axis here) at $(0, \pm a) = (0, \pm 2)$.

The foci are at $(0, \pm c) = (0, \pm \sqrt{13})$.

The eccentricity is $e = \frac{c}{a} = \frac{\sqrt{13}}{2}$.

The length of the latus rectum is $\frac{2b^2}{a} = \frac{2(9)}{2} = 9$.

The equation $9y^2 - 4x^2 = 36$ describes a hyperbola, but not in its most useful form. To extract geometric information—where the vertices and foci sit, how "stretched" the hyperbola is—we need the standard form. For a hyperbola, this means getting $1$ on the right-hand side and identifying which variable is positive (that determines the axis).

Dividing the entire equation by $36$:

This is now in standard form. Because the $y^2$ term is positive, the hyperbola opens vertically (along the $y$-axis). The standard template for such a hyperbola centered at the origin is:

where $a^2 = 4$ and $b^2 = 9$.

Now we extract each piece systematically.

Finding the vertices

The vertices lie on the transverse axis at a distance $a$ from the center. Here $a^2 = 4$, so $a = 2$.

Vertices: $(0, 2)$ and $(0, -2)$.

Finding the foci

The foci lie further out along the same axis. The relationship for hyperbolas is $c^2 = a^2 + b^2$ (note the plus sign, unlike ellipses).

Foci: $(0, \sqrt{13})$ and $(0, -\sqrt{13})$.

Finding the eccentricity

Eccentricity measures how "open" the hyperbola is. For any hyperbola, $e > 1$.

This value is approximately $1.8$, confirming the hyperbola is moderately open.

Finding the length of the latus rectum

The latus rectum is the chord through a focus, perpendicular to the transverse axis. Its length is given by: