Find the equation of the hyperbola with :
Foci `(0,pm sqrt(10))`, pasing through (2,3).

Answer

To find the equation of the hyperbola with foci at (0, ±√10) and passing through the point (2, 3), we can follow these steps:

### Step 1: Identify the type of hyperbola
Given that the foci are at (0, ±√10), we know that this is a vertical hyperbola. The general equation for a vertical hyperbola is:

\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]

### Step 2: Determine the relationship between a, b, and the foci
The distance from the center to the foci (c) is given by:

\[ c = \sqrt{a^2 + b^2} \]

From the foci, we have:

\[ c = \sqrt{10} \]

Thus, we can write:

\[ c^2 = 10 \implies a^2 + b^2 = 10 \quad \text{(1)} \]

### Step 3: Use the point (2, 3) to form another equation
Since the hyperbola passes through the point (2, 3), we can substitute these values into the hyperbola equation:

\[ \frac{3^2}{a^2} - \frac{2^2}{b^2} = 1 \]

This simplifies to:

\[ \frac{9}{a^2} - \frac{4}{b^2} = 1 \quad \text{(2)} \]

### Step 4: Solve the system of equations
From equation (1):

\[ b^2 = 10 - a^2 \]

Substituting this into equation (2):

\[ \frac{9}{a^2} - \frac{4}{10 - a^2} = 1 \]

### Step 5: Clear the fractions
Multiply through by \(a^2(10 - a^2)\):

\[ 9(10 - a^2) - 4a^2 = a^2(10 - a^2) \]

Expanding this gives:

\[ 90 - 9a^2 - 4a^2 = 10a^2 - a^4 \]

Rearranging terms leads to:

\[ a^4 - 23a^2 + 90 = 0 \]

### Step 6: Let \(u = a^2\)
This gives us a quadratic equation:

\[ u^2 - 23u + 90 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula:

\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{23 \pm \sqrt{(-23)^2 - 4 \cdot 1 \cdot 90}}{2 \cdot 1} \]

Calculating the discriminant:

\[ 23^2 - 360 = 529 - 360 = 169 \]

Thus:

\[ u = \frac{23 \pm 13}{2} \]

Calculating the two possible values:

1. \(u = \frac{36}{2} = 18\)
2. \(u = \frac{10}{2} = 5\)

### Step 8: Find \(a^2\) and \(b^2\)
1. If \(a^2 = 18\), then \(b^2 = 10 - 18 = -8\) (not possible).
2. If \(a^2 = 5\), then \(b^2 = 10 - 5 = 5\).

### Step 9: Write the equation of the hyperbola
Substituting \(a^2\) and \(b^2\) into the hyperbola equation:

\[ \frac{y^2}{5} - \frac{x^2}{5} = 1 \]

This can be simplified to:

\[ \frac{y^2 - x^2}{5} = 1 \implies y^2 - x^2 = 5 \]

### Final Equation
The equation of the hyperbola is:

\[ y^2 - x^2 = 5 \]