Q.Find the equation of the hyperbola satisfying the given conditions: Foci $(0, \pm \sqrt{10})$, passing through $(2, 3)$.

The key idea is that the foci lie on the $y$-axis, so the hyperbola is vertical. Its standard form is

with $c^2 = a^2 + b^2$ and $c = \sqrt{10}$.

Step 1: From $c = \sqrt{10}$, we have $a^2 + b^2 = 10$.

Step 2: The point $(2, 3)$ lies on the hyperbola, so

Step 3: Substitute $b^2 = 10 - a^2$ into the equation:

Multiply through by $a^2(10 - a^2)$:

$9(10 - a^2) - 4a^2 = a^2(10 - a^2).$ …

Concept first: Why this approach works

When a hyperbola’s foci lie on the y-axis at $(0, \pm c)$, the transverse axis is vertical. The standard form for such a hyperbola centered at the origin is:

Here, $c$ is the distance from the center to each focus, and the fundamental relation for a hyperbola is $c^2 = a^2 + b^2$ (note: plus, not minus — that’s for ellipses). The given point $(2,3)$ lies on the curve, so it must satisfy the equation. We have two unknowns ($a^2$ and $b^2$) and two conditions: the foci give $c^2$, and the point gives a second equation.

Step-by-step solution

1. Identify $c$ from the foci.

   The foci are $(0, \pm \sqrt{10})$, so $c = \sqrt{10}$ and therefore $c^2 = 10$.

2. Write the standard equation.

   Since the foci are on the y-axis, the hyperbola opens upward and downward:

The relation between $a$, $b$, and $c$ is:

3. Use the given point $(2,3)$. Substitute $x = 2$, $y = 3$ into the equation:

4. Solve the system of equations. From (1), $b^2 = 10 - a^2$. Substitute into (2):

Multiply through by $a^2(10 - a^2)$ to clear denominators:

Simplify:

Bring all terms to one side: …