If the length of the eccentricity of an ellipse is `(3)/(8)` and the distance between the foci is 6 units, then the length of the latus rectum of the ellipse is

Options

A. (a)`(55)/(8)`
B. (b)`(55)/(4)`
C. (c)`(8)/(55)`
D. (d)`(4)/(55)`

Answer

To solve the problem step by step, we will follow the information given and apply the relevant formulas for an ellipse.

### Step 1: Understand the given information
- Eccentricity \( e = \frac{3}{8} \)
- Distance between the foci \( 2ae = 6 \)

### Step 2: Calculate the semi-major axis \( a \)
From the distance between the foci, we have:
\[ 2ae = 6 \]
Substituting the value of \( e \):
\[ 2a \left(\frac{3}{8}\right) = 6 \]
Now, simplify this:
\[ \frac{3a}{4} = 6 \]
To isolate \( a \), multiply both sides by \( \frac{4}{3} \):
\[ a = 6 \cdot \frac{4}{3} = 8 \]

### Step 3: Calculate \( b^2 \) using the formula
We know that:
\[ b^2 = a^2(1 - e^2) \]
First, calculate \( e^2 \):
\[ e^2 = \left(\frac{3}{8}\right)^2 = \frac{9}{64} \]
Now, calculate \( 1 - e^2 \):
\[ 1 - e^2 = 1 - \frac{9}{64} = \frac{64 - 9}{64} = \frac{55}{64} \]
Now substitute \( a = 8 \) into the equation for \( b^2 \):
\[ b^2 = 8^2 \left(\frac{55}{64}\right) = 64 \cdot \frac{55}{64} = 55 \]

### Step 4: Calculate the length of the latus rectum
The length of the latus rectum \( L \) of an ellipse is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \( b^2 \) and \( a \):
\[ L = \frac{2 \cdot 55}{8} = \frac{110}{8} = \frac{55}{4} \]

### Final Answer
The length of the latus rectum of the ellipse is \( \frac{55}{4} \).

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