Find the equation of the hyperbola whose foci are `(0, +-6)` and conjugate axis is `2sqrt(11)`.

Answer

To find the equation of the hyperbola with given foci and conjugate axis, we can follow these steps:

### Step 1: Identify the given information
The foci of the hyperbola are given as (0, ±6), which indicates that the hyperbola opens vertically. The conjugate axis is given as \(2\sqrt{11}\).

### Step 2: Write the standard form of the hyperbola
Since the foci are on the y-axis, the standard form of the equation of the hyperbola is:
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]

### Step 3: Determine the value of \(c\)
The coordinates of the foci are (0, ±c), where \(c = 6\). Thus, we have:
\[ c = 6 \]

### Step 4: Relate the conjugate axis to \(b\)
The conjugate axis is given as \(2b\). From the problem, we know:
\[ 2b = 2\sqrt{11} \]
Dividing both sides by 2 gives:
\[ b = \sqrt{11} \]

### Step 5: Use the relationship between \(a\), \(b\), and \(c\)
We know the relationship:
\[ c^2 = a^2 + b^2 \]
Substituting the known values:
\[ 6^2 = a^2 + (\sqrt{11})^2 \]
This simplifies to:
\[ 36 = a^2 + 11 \]

### Step 6: Solve for \(a^2\)
Rearranging the equation gives:
\[ a^2 = 36 - 11 = 25 \]

### Step 7: Write the final equation of the hyperbola
Now that we have \(a^2\) and \(b^2\), we can substitute these values back into the standard form of the hyperbola:
\[ \frac{y^2}{25} - \frac{x^2}{11} = 1 \]

Thus, the equation of the hyperbola is:
\[ \frac{y^2}{25} - \frac{x^2}{11} = 1 \]

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