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

Options

A. `2sqrt(2)`
B. `4`
C. `4sqrt(2)`
D. `2`

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 of the ellipse:

\[ 5x^2 + 9y^2 = 45 \]

To convert it into 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 the ellipse is given by the formula:

\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

Substituting the values of \(a^2\) and \(b^2\):

\[ e = \sqrt{1 - \frac{5}{9}} = \sqrt{\frac{9 - 5}{9}} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]

### 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 \(e\):

\[ \text{Distance} = 2 \times 3 \times \frac{2}{3} = 2 \times 2 = 4 \]

### Final Answer
The distance between the foci of the ellipse is:

\[ \boxed{4} \]

---