Q.Find the equation of the hyperbola satisfying the given conditions: Vertices $(\pm 2, 0)$, foci $(\pm 3, 0)$.

The vertices and foci both lie on the $x$-axis, so this is a horizontal hyperbola with center at the origin. Its standard form is:

Step 1: The vertices $(\pm 2, 0)$ give $a = 2$, so $a^2 = 4$.

Step 2: The foci $(\pm 3, 0)$ give $c = 3$, so $c^2 = 9$.

Step 3: Use the hyperbola relation $c^2 = a^2 + b^2$:

$9 = 4 + b^2 \implies b^2 = 5$ …

Why this approach works

A hyperbola is the locus of points where the absolute difference of distances to two foci is constant. The standard forms depend on which axis the foci lie along. When both vertices and foci sit on the $x$-axis (as they do here), the transverse axis is horizontal and the equation takes the form

where $a$ is the distance from the center to each vertex, $c$ is the distance from the center to each focus, and these are related by $c^2 = a^2 + b^2$.

The vertices tell us $a$ directly. The foci give us $c$. The relationship then unlocks $b^2$.

Solution

1. Identify the center and orientation.

   The vertices $(\pm 2, 0)$ and foci $(\pm 3, 0)$ are symmetric about the origin, so the center is at $(0, 0)$. Both lie on the $x$-axis, confirming a horizontal transverse axis.

2. Read off $a$ from the vertices.

   The vertices are at $(\pm a, 0)$, so comparing with $(\pm 2, 0)$ gives

3. Read off $c$ from the foci. The foci are at $(\pm c, 0)$, so from $(\pm 3, 0)$ we have

4. Use the fundamental relation to find $b^2$. …