The radius of the auxiliary circle of the hyperbola `x^(2)/25-y^(2)/9=1` is

Options

A. 3
B. 4
C. 5
D. 6

Answer

To find the radius of the auxiliary circle of the hyperbola given by the equation \( \frac{x^2}{25} - \frac{y^2}{9} = 1 \), we can follow these steps:

### Step 1: Identify the standard form of the hyperbola
The standard form of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
From the given equation \( \frac{x^2}{25} - \frac{y^2}{9} = 1 \), we can identify:
- \( a^2 = 25 \)
- \( b^2 = 9 \)

### Step 2: Calculate the values of \( a \) and \( b \)
To find \( a \) and \( b \), we take the square roots:
\[ a = \sqrt{25} = 5 \]
\[ b = \sqrt{9} = 3 \]

### Step 3: Determine the radius of the auxiliary circle
The radius of the auxiliary circle of a hyperbola is equal to \( a \). Therefore, we have:
\[ \text{Radius of the auxiliary circle} = a = 5 \]

### Conclusion
Thus, the radius of the auxiliary circle of the hyperbola \( \frac{x^2}{25} - \frac{y^2}{9} = 1 \) is \( 5 \).