If the eccentricity of an ellipse is `1/sqrt2`, then its latusrectum is equal to its

Options

A. minor axis
B. semi minor axis
C. major axis
D. semi major axis

Answer

To solve the problem, we need to find the relationship between the eccentricity of the ellipse and its latus rectum. Given that the eccentricity \( e \) of the ellipse is \( \frac{1}{\sqrt{2}} \), we will follow these steps:

### Step 1: Use the formula for eccentricity
The formula for the eccentricity \( e \) of an ellipse is given by:

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

where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

### Step 2: Substitute the given eccentricity
We substitute \( e = \frac{1}{\sqrt{2}} \) into the formula:

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

### Step 3: Square both sides
Squaring both sides to eliminate the square root gives:

\[ \left(\frac{1}{\sqrt{2}}\right)^2 = 1 - \frac{b^2}{a^2} \]

This simplifies to:

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

### Step 4: Rearrange the equation
Rearranging the equation to isolate \( \frac{b^2}{a^2} \):

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

### Step 5: Express \( b^2 \) in terms of \( a^2 \)
Now, we can express \( b^2 \):

\[ b^2 = \frac{a^2}{2} \]

### Step 6: Use the formula for the latus rectum
The formula for the length of the latus rectum \( L \) of an ellipse is given by:

\[ L = \frac{2b^2}{a} \]

### Step 7: Substitute \( b^2 \) into the latus rectum formula
Substituting \( b^2 = \frac{a^2}{2} \) into the latus rectum formula:

\[ L = \frac{2 \cdot \frac{a^2}{2}}{a} \]

This simplifies to:

\[ L = \frac{a^2}{a} = a \]

### Step 8: Conclusion
Thus, the length of the latus rectum \( L \) is equal to the semi-major axis \( a \).

### Final Answer
The latus rectum is equal to the semi-major axis.

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