The eccentricity of the ellipse `9x^2+25y^2=225` is e then the value of 5e is

Options

A. 3
B. 4
C. 2
D. 1

Answer

To find the value of \(5e\) where \(e\) is the eccentricity of the ellipse given by the equation \(9x^2 + 25y^2 = 225\), we can follow these steps:

### Step 1: Write the equation in standard form
The standard form of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
To convert the given equation \(9x^2 + 25y^2 = 225\) into this form, we divide the entire equation by 225.

\[ \frac{9x^2}{225} + \frac{25y^2}{225} = 1 \]

### Step 2: Simplify the fractions
Now simplify the fractions:
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
This gives us \(a^2 = 25\) and \(b^2 = 9\).

### Step 3: Identify the values of \(a\) and \(b\)
From \(a^2 = 25\) and \(b^2 = 9\), we find:
\[ a = \sqrt{25} = 5 \quad \text{and} \quad b = \sqrt{9} = 3 \]

### Step 4: Calculate the eccentricity \(e\)
The formula for the eccentricity \(e\) of an ellipse is given by:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values of \(b^2\) and \(a^2\):
\[ e = \sqrt{1 - \frac{9}{25}} \]

### Step 5: Simplify the expression under the square root
To simplify:
\[ e = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{25 - 9}{25}} = \sqrt{\frac{16}{25}} \]

### Step 6: Calculate \(e\)
Taking the square root:
\[ e = \frac{4}{5} \]

### Step 7: Find \(5e\)
Now, we need to find \(5e\):
\[ 5e = 5 \times \frac{4}{5} = 4 \]

### Final Answer
Thus, the value of \(5e\) is \(4\).

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