Find the equation of hyperbola whose transverse axis is 10 conjugate axis is 6.

Answer

To find the equation of the hyperbola with a transverse axis of 10 and a conjugate axis of 6, we can follow these steps:

### Step 1: Identify the lengths of the axes
The transverse axis length is given as 10, and the conjugate axis length is given as 6.

### Step 2: Relate the lengths to 'a' and 'b'
The length of the transverse axis is \(2a\) and the length of the conjugate axis is \(2b\). Therefore, we can set up the following equations:
- \(2a = 10\)
- \(2b = 6\)

### Step 3: Solve for 'a' and 'b'
From the equation \(2a = 10\):
\[ a = \frac{10}{2} = 5 \]

From the equation \(2b = 6\):
\[ b = \frac{6}{2} = 3 \]

### Step 4: Write the standard form of the hyperbola
The standard form of a hyperbola that opens horizontally is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

### Step 5: Substitute the values of 'a' and 'b'
Now, substitute \(a = 5\) and \(b = 3\) into the equation:
\[ \frac{x^2}{5^2} - \frac{y^2}{3^2} = 1 \]

### Step 6: Simplify the equation
Calculating \(5^2\) and \(3^2\):
\[ \frac{x^2}{25} - \frac{y^2}{9} = 1 \]

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

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