Q.Find the equation of the hyperbola satisfying the given conditions: Vertices $(0, \pm 3)$, foci $(0, \pm 5)$.

Concept: Hyperbola Standard Form (Vertical Transverse Axis)

Since the vertices and foci lie on the $y$-axis, the hyperbola is of the form

Step 1: Identify $a$ and $c$

Vertices $(0, \pm 3)$ give $a = 3$. Foci $(0, \pm 5)$ give $c = 5$.

Step 2: Find $b^2$ using $c^2 = a^2 + b^2$

Step 3: Write the equation …

The key to this problem is reading the coordinates carefully. The vertices are at $(0, \pm 3)$ and the foci at $(0, \pm 5)$. Notice that both the vertices and foci lie on the y-axis. This tells us the hyperbola opens upward and downward — its transverse axis is vertical.

For a hyperbola centred at the origin with a vertical transverse axis, the standard form is:

Here, $a$ is the distance from the centre to each vertex, and $c$ is the distance from the centre to each focus. The relationship between $a$, $b$, and $c$ for a hyperbola is $c^2 = a^2 + b^2$.

Let’s work through it step by step.

1. Identify $a$ and $c$ from the given points.

   The vertices are $(0, \pm 3)$, so the distance from the centre $(0,0)$ to a vertex is $a = 3$.

   The foci are $(0, \pm 5)$, so the distance from the centre to a focus is $c = 5$.

2. Find $b^2$ using the hyperbola relation.

   For any hyperbola, $c^2 = a^2 + b^2$.

   Substitute: $5^2 = 3^2 + b^2$

   $25 = 9 + b^2$

   $b^2 = 16$.

   Watch out

   A common mistake is to use $c^2 = a^2 - b^2$ (which is for ellipses). For hyperbolas, it’s always $c^2 = a^2 + b^2$.

3. Write the equation in standard form.

   Since the transverse axis is vertical, $y^2$ comes first. …