What is the length of the major axis of the ellipse whose equation is `10x^(2)+20y^(2)=200?`

Options

A. 3.16
B. 4.47
C. 6.32
D. 8.94

Explanation

To find the length of the major axis of the ellipse given by the equation \(10x^2 + 20y^2 = 200\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the ellipse:
\[ 10x^2 + 20y^2 = 200 \]
To convert this into the standard form of an ellipse, we divide the entire equation by 200:
\[ \frac{10x^2}{200} + \frac{20y^2}{200} = 1 \]
This simplifies to:
\[ \frac{x^2}{20} + \frac{y^2}{10} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
In the standard form of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 20\)
- \(b^2 = 10\)

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

### Step 4: Find the length of the major axis
The length of the major axis is given by \(2a\):
\[ \text{Length of the major axis} = 2a = 2 \times 2\sqrt{5} = 4\sqrt{5} \]

### Conclusion
Thus, the length of the major axis of the ellipse is:
\[ \boxed{4\sqrt{5}} \]

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