Find the eccentricity of the ellipse whose latus rectum is one third of the major axis.

Answer

To find the eccentricity of the ellipse whose latus rectum is one third of the major axis, we can follow these steps:

### Step 1: Understand the relationship between the latus rectum and the major axis
The length of the latus rectum (L) of an ellipse is given by the formula:
\[ L = \frac{2B^2}{A} \]
where \(A\) is the semi-major axis and \(B\) is the semi-minor axis.

### Step 2: Express the major axis
The length of the major axis is given by:
\[ \text{Major Axis} = 2A \]
According to the problem, the latus rectum is one third of the major axis:
\[ L = \frac{1}{3} \times 2A = \frac{2A}{3} \]

### Step 3: Set up the equation
From the above relationships, we can equate the two expressions for the latus rectum:
\[ \frac{2B^2}{A} = \frac{2A}{3} \]

### Step 4: Simplify the equation
We can simplify this equation by multiplying both sides by \(A\):
\[ 2B^2 = \frac{2A^2}{3} \]
Now, divide both sides by 2:
\[ B^2 = \frac{A^2}{3} \]

### Step 5: Find the relationship between \(B^2\) and \(A^2\)
We can express \(B^2\) in terms of \(A^2\):
\[ \frac{B^2}{A^2} = \frac{1}{3} \]

### Step 6: Use the eccentricity formula
The eccentricity \(e\) of the ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{B^2}{A^2}} \]
Substituting the value we found:
\[ e = \sqrt{1 - \frac{1}{3}} = \sqrt{\frac{2}{3}} \]

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