What is the area of he ellipse `4x^(2)+9y^(2)=1`

Options

A. `6pi`
B. `(pi)/36`
C. `(pi)/6`
D. `(pi)/(sqrt(6))`

Answer

To find the area of the ellipse given by the equation \(4x^2 + 9y^2 = 1\), we can follow these steps:

### Step 1: Rewrite the equation in standard form
The standard form of an ellipse is given by the equation:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
We need to rewrite the given equation \(4x^2 + 9y^2 = 1\) in this form.

### Step 2: Divide the entire equation by 1
To convert the equation into the standard form, we can divide each term by 1:
\[ \frac{4x^2}{1} + \frac{9y^2}{1} = 1 \]
This can be rewritten as:
\[ \frac{x^2}{\frac{1}{4}} + \frac{y^2}{\frac{1}{9}} = 1 \]

### Step 3: Identify \(a^2\) and \(b^2\)
From the standard form, we can identify:
- \(a^2 = \frac{1}{4}\) → \(a = \frac{1}{2}\)
- \(b^2 = \frac{1}{9}\) → \(b = \frac{1}{3}\)

### Step 4: Use the formula for the area of the ellipse
The area \(A\) of an ellipse is given by the formula:
\[ A = \pi \cdot a \cdot b \]
Substituting the values of \(a\) and \(b\):
\[ A = \pi \cdot \frac{1}{2} \cdot \frac{1}{3} \]

### Step 5: Calculate the area
Now, we can calculate the area:
\[ A = \pi \cdot \frac{1}{2} \cdot \frac{1}{3} = \frac{\pi}{6} \]

### Final Answer
Thus, the area of the ellipse \(4x^2 + 9y^2 = 1\) is:
\[ \frac{\pi}{6} \]

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