Find the equation of the hyperbola whose transverse and conjugate axes are the x and y axes respectively, given that the length of conjugate axis is 5 and distance between the foci is 13.

Answer

To find the equation of the hyperbola whose transverse and conjugate axes are the x and y axes respectively, given that the length of the conjugate axis is 5 and the distance between the foci is 13, we can follow these steps:

### Step 1: Identify the standard form of the hyperbola
The standard equation of a hyperbola with transverse axis along the x-axis and conjugate axis along the y-axis is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

### Step 2: Use the length of the conjugate axis
The length of the conjugate axis is given as 5. The length of the conjugate axis is represented by \(2b\):
\[ 2b = 5 \implies b = \frac{5}{2} \]

### Step 3: Use the distance between the foci
The distance between the foci is given as 13. The distance between the foci is represented by \(2ae\), where \(e\) is the eccentricity. Therefore:
\[ 2ae = 13 \implies ae = \frac{13}{2} \]

### Step 4: Relate \(a\), \(b\), and \(e\)
We know the relationship between \(a\), \(b\), and \(e\) is given by:
\[ e^2 = 1 + \frac{b^2}{a^2} \]
Thus, we can express \(e\) in terms of \(a\) and \(b\):
\[ e = \frac{c}{a} \quad \text{where } c = \sqrt{a^2 + b^2} \]

### Step 5: Calculate \(b^2\)
Now, we can calculate \(b^2\):
\[ b = \frac{5}{2} \implies b^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]

### Step 6: Substitute \(e\) in terms of \(a\)
From \(ae = \frac{13}{2}\), we can express \(e\) as:
\[ e = \frac{13}{2a} \]

### Step 7: Substitute \(e\) in the eccentricity formula
Substituting \(e\) into the eccentricity formula:
\[ \left(\frac{13}{2a}\right)^2 = 1 + \frac{\frac{25}{4}}{a^2} \]
This simplifies to:
\[ \frac{169}{4a^2} = 1 + \frac{25}{4a^2} \]

### Step 8: Clear the fractions
Multiply through by \(4a^2\) to eliminate the denominators:
\[ 169 = 4a^2 + 25 \]

### Step 9: Solve for \(a^2\)
Rearranging gives:
\[ 4a^2 = 169 - 25 = 144 \implies a^2 = \frac{144}{4} = 36 \]

### Step 10: Write the equation of the hyperbola
Now that we have \(a^2\) and \(b^2\):
\[ a^2 = 36 \quad \text{and} \quad b^2 = \frac{25}{4} \]
Substituting these values into the standard form of the hyperbola:
\[ \frac{x^2}{36} - \frac{y^2}{\frac{25}{4}} = 1 \]
Multiplying through by 4 to eliminate the fraction:
\[ \frac{4x^2}{36} - \frac{y^2}{25} = 1 \implies \frac{x^2}{9} - \frac{y^2}{25} = 1 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ \frac{x^2}{9} - \frac{y^2}{25} = 1 \]