Q.Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 5, 0)$, the transverse axis is of length $8$.

Concept: Standard form of a hyperbola with horizontal transverse axis.

Since the foci are at $(\pm 5, 0)$ on the $x$-axis, the hyperbola has its center at the origin with a horizontal transverse axis. The standard form is:

Step 1: The transverse axis has length $2a = 8$, so $a = 4$ and $a^2 = 16$.

Step 2: The foci are at $(\pm c, 0)$ where $c = 5$. …

The foci lie on the $x$-axis and are symmetric about the origin, which tells us immediately that this is a horizontal hyperbola centered at the origin. The standard form for such a hyperbola is

where the vertices are at $(\pm a, 0)$ and the foci at $(\pm c, 0)$. The relationship between these parameters is the fundamental identity $c^2 = a^2 + b^2$ for hyperbolas—notice the plus sign, which distinguishes it from the ellipse.

The transverse axis is the segment joining the two vertices, so its length is $2a$. The distance from center to focus is $c$. Our task is to extract $a$ and $c$ from the given information, then use the identity to find $b^2$.

1. Find $a$ from the transverse axis length.

   The transverse axis has length $8$, so:

2. Identify $c$ from the foci.

   The foci are at $(\pm 5, 0)$, which means:

3. Calculate $b^2$ using the hyperbola identity.

   We know $c^2 = a^2 + b^2$, so:

$b^2 = 9$ …