The length of the transverse axis of the hyperbola `x^(2) -20y^(2) = 20` is

Options

A. 1
B. `2sqrt(5)`
C. `4sqrt(5)`
D. 2

Answer

To find the length of the transverse axis of the hyperbola given by the equation \( x^2 - 20y^2 = 20 \), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the hyperbola:
\[ x^2 - 20y^2 = 20 \]
To convert this into standard form, we divide every term by 20:
\[ \frac{x^2}{20} - \frac{20y^2}{20} = \frac{20}{20} \]
This simplifies to:
\[ \frac{x^2}{20} - \frac{y^2}{1} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form of the hyperbola:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
we can identify:
\[ a^2 = 20 \quad \text{and} \quad b^2 = 1 \]

### Step 3: Calculate \(a\)
To find \(a\), we take the square root of \(a^2\):
\[ a = \sqrt{20} = 2\sqrt{5} \]

### Step 4: Calculate the length of the transverse axis
The length of the transverse axis of a hyperbola is given by:
\[ \text{Length of transverse axis} = 2a \]
Substituting the value of \(a\):
\[ \text{Length of transverse axis} = 2 \times 2\sqrt{5} = 4\sqrt{5} \]

### Final Answer
Thus, the length of the transverse axis of the hyperbola is:
\[ \boxed{4\sqrt{5}} \]

---