The co-ordinates of the foci of the ellipse `9x^(2) + 4y^(2) = 36 ` are :

Options

A. `(0, pm sqrt(5)) `
B. `(pm 5, 0 ) `
C. (0,0)
D. none of these.

Answer

To find the coordinates of the foci of the ellipse given by the equation \(9x^2 + 4y^2 = 36\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation:
\[ 9x^2 + 4y^2 = 36 \]
To convert this into standard form, we divide every term by 36:
\[ \frac{9x^2}{36} + \frac{4y^2}{36} = 1 \]
This simplifies to:
\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]

### 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:
- \(a^2 = 4\) (so \(a = 2\))
- \(b^2 = 9\) (so \(b = 3\))

### Step 3: Determine the orientation of the ellipse
Since \(b^2 > a^2\) (i.e., \(9 > 4\)), the major axis is along the y-axis.

### Step 4: Calculate the distance to the foci
The distance \(c\) from the center to each focus is given by the formula:
\[ c = \sqrt{b^2 - a^2} \]
Substituting the values we found:
\[ c = \sqrt{9 - 4} = \sqrt{5} \]

### Step 5: Find the coordinates of the foci
Since the major axis is along the y-axis, the coordinates of the foci are:
\[ (0, \pm c) = (0, \pm \sqrt{5}) \]

### Final Answer
The coordinates of the foci of the ellipse are:
\[ (0, \sqrt{5}) \quad \text{and} \quad (0, -\sqrt{5}) \]

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