Find the equation of hyperbola where, conjugate axis is 3 along Y-axis and distance between foci is 5.

Options

A. `(4x^(2))/(9) - (y^(2))/(4) = 1`
B. `(x^(2))/(4) - (4y^(2))/(9) = 1`
C. `(x^(2))/(4) - (y^(2))/(9) = 1`
D. `4x^(2) - 36y^(2) = 9`

Answer

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

### Step 1: Identify the given information
- The length of the conjugate axis is 3 along the Y-axis.
- The distance between the foci is 5.

### Step 2: Determine the values of \(c\) and \(b\)
- The distance between the foci is given by \(2c\). Therefore, we can write:
\[ 2c = 5 \implies c = \frac{5}{2} \]

- The length of the conjugate axis is given by \(2b\). Thus:
\[ 2b = 3 \implies b = \frac{3}{2} \]

### Step 3: Use the relationship between \(a\), \(b\), and \(c\)
For hyperbolas, the relationship between \(a\), \(b\), and \(c\) is given by:
\[ c^2 = a^2 + b^2 \]
Substituting the values of \(c\) and \(b\):
\[ \left(\frac{5}{2}\right)^2 = a^2 + \left(\frac{3}{2}\right)^2 \]
Calculating \(c^2\) and \(b^2\):
\[ \frac{25}{4} = a^2 + \frac{9}{4} \]

### Step 4: Solve for \(a^2\)
Rearranging the equation:
\[ a^2 = \frac{25}{4} - \frac{9}{4} = \frac{16}{4} = 4 \]
Thus, we find:
\[ a = 2 \]

### Step 5: Write the equation of the hyperbola
Since the conjugate axis is along the Y-axis, the standard form of the hyperbola is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Substituting the values of \(a^2\) and \(b^2\):
\[ \frac{x^2}{4} - \frac{y^2}{\left(\frac{3}{2}\right)^2} = 1 \]
This simplifies to:
\[ \frac{x^2}{4} - \frac{4y^2}{9} = 1 \]

### Final Equation
The final equation of the hyperbola is:
\[ \frac{x^2}{4} - \frac{4y^2}{9} = 1 \]

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