If the length of the major axis of an ellipse in 3 times the length of minor axis , then its eccentricity is

Options

A. `(2)/(3)`
B. `(sqrt(2))/(3)`
C. `(2 sqrt(2))/(3)`
D. `(1)/(3)`

Answer

To find the eccentricity of an ellipse where the length of the major axis is three times the length of the minor axis, we can follow these steps:

### Step 1: Understand the relationship between the axes of the ellipse.
The lengths of the major and minor axes of an ellipse are given by:
- Length of the major axis = \(2a\)
- Length of the minor axis = \(2b\)

According to the problem, we have:
\[ 2a = 3 \times 2b \]

### Step 2: Simplify the equation.
We can simplify the equation by dividing both sides by 2:
\[ a = 3b \]

### Step 3: Square both sides to find a relationship between \(a^2\) and \(b^2\).
Squaring both sides gives:
\[ a^2 = (3b)^2 \]
\[ a^2 = 9b^2 \]

### Step 4: Find the ratio of \(b^2\) to \(a^2\).
From the equation \(a^2 = 9b^2\), we can express \(b^2\) in terms of \(a^2\):
\[ \frac{b^2}{a^2} = \frac{1}{9} \]

### Step 5: Use the formula for eccentricity.
The eccentricity \(e\) of an ellipse is given by the formula:
\[ e^2 = 1 - \frac{b^2}{a^2} \]
Substituting the ratio we found:
\[ e^2 = 1 - \frac{1}{9} \]
\[ e^2 = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \]

### Step 6: Take the square root to find \(e\).
Taking the square root of both sides gives:
\[ e = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3} \]

### Final Answer:
The eccentricity of the ellipse is:
\[ e = \frac{2\sqrt{2}}{3} \]

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