The distance between the foci of the ellipse `5x^(2)+9y^(2)=45` is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the distance between the foci of the ellipse given by the equation \(5x^2 + 9y^2 = 45\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation:
\[ 5x^2 + 9y^2 = 45 \]
To rewrite this in standard form, we divide the entire equation by 45:
\[ \frac{5x^2}{45} + \frac{9y^2}{45} = 1 \]
This simplifies to:
\[ \frac{x^2}{9} + \frac{y^2}{5} = 1 \]

### 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:
\[ a^2 = 9 \quad \text{and} \quad b^2 = 5 \]
Thus, we find:
\[ a = 3 \quad \text{and} \quad b = \sqrt{5} \]

### Step 3: Determine the eccentricity \(e\)
The eccentricity \(e\) of an ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values we found:
\[ e = \sqrt{1 - \frac{5}{9}} = \sqrt{\frac{9 - 5}{9}} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]

### Step 4: Calculate the distance between the foci
The distance between the foci of the ellipse is given by the formula:
\[ \text{Distance} = 2ae \]
Substituting the values of \(a\) and \(e\):
\[ \text{Distance} = 2 \cdot 3 \cdot \frac{2}{3} \]
This simplifies to:
\[ \text{Distance} = 2 \cdot 2 = 4 \]

### Final Answer
The distance between the foci of the ellipse is \(4\).

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