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 latus rectum and 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: Set up the equation based on the problem statement
According to the problem, the latus rectum is one third of the major axis. The length of the major axis is \(2a\), so we can write:
\[ \frac{2b^2}{a} = \frac{1}{3} \times 2a \]
This simplifies to:
\[ \frac{2b^2}{a} = \frac{2}{3}a \]

### Step 3: Simplify the equation
We can cancel the 2 from both sides:
\[ \frac{b^2}{a} = \frac{1}{3}a \]
Multiplying both sides by \(a\) gives:
\[ b^2 = \frac{1}{3}a^2 \]

### Step 4: Use the relationship between eccentricity, \(e\), and \(a\) and \(b\)
The eccentricity \(e\) of the ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

### Step 5: Substitute \(b^2\) into the eccentricity formula
Now, substituting \(b^2 = \frac{1}{3}a^2\) into the eccentricity formula:
\[ e = \sqrt{1 - \frac{\frac{1}{3}a^2}{a^2}} \]
This simplifies to:
\[ e = \sqrt{1 - \frac{1}{3}} = \sqrt{\frac{2}{3}} \]

### Step 6: Final result
Thus, the eccentricity of the ellipse is:
\[ e = \sqrt{\frac{2}{3}} \]