The foci of the ellipse `36x^(2) + 9y^(2) = 324` are

Options

A. `0,pm3sqrt(3)`
B. `0,pm sqrt(3)`
C. `0,pm4sqrt(3)`
D. `0,pm5sqrt(3)`

Answer

To find the foci of the ellipse given by the equation \(36x^2 + 9y^2 = 324\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the ellipse:
\[ 36x^2 + 9y^2 = 324 \]
To convert this into standard form, we divide both sides by 324:
\[ \frac{36x^2}{324} + \frac{9y^2}{324} = 1 \]
This simplifies to:
\[ \frac{x^2}{9} + \frac{y^2}{36} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
\[ a^2 = 9 \quad \text{and} \quad b^2 = 36 \]
Thus, we have:
\[ a = 3 \quad \text{and} \quad b = 6 \]

### Step 3: Determine the orientation of the ellipse
Since \(b^2 > a^2\) (i.e., \(36 > 9\)), the major axis is along the y-axis.

### Step 4: Calculate the foci
The foci of an ellipse are given by the coordinates \((0, \pm c)\), where \(c\) is calculated using the formula:
\[ c = \sqrt{b^2 - a^2} \]
Substituting the values of \(b^2\) and \(a^2\):
\[ c = \sqrt{36 - 9} = \sqrt{27} = 3\sqrt{3} \]

### Step 5: Write the coordinates of the foci
Thus, the coordinates of the foci are:
\[ (0, \pm 3\sqrt{3}) \]

### Final Answer
The foci of the ellipse are:
\[ (0, 3\sqrt{3}) \quad \text{and} \quad (0, -3\sqrt{3}) \]

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