If the eccentricity of and ellipse is `(5)/(8)` and the distance between its foci is 10, then length of its latus rectum is

Options

A. (a) `(39)/(4)`
B. (b) `(39)/(2)`
C. (c) `(23)/(2)`
D. (d) `(23)/(4)`

Answer

To solve the problem step by step, we will follow the process outlined in the video transcript.

### Step 1: Understand the given information
We are given:
- Eccentricity \( e = \frac{5}{8} \)
- Distance between the foci \( 2c = 10 \)

### Step 2: Find the value of \( c \)
The distance between the foci is represented as \( 2c \), so we can find \( c \):
\[ c = \frac{10}{2} = 5 \]

### Step 3: Relate \( c \), \( a \), and \( e \)
For an ellipse, the relationship between \( a \), \( b \), and \( c \) is given by:
\[ c = ae \]
Substituting the values we have:
\[ 5 = a \cdot \frac{5}{8} \]

### Step 4: Solve for \( a \)
To find \( a \), we rearrange the equation:
\[ a = \frac{5 \cdot 8}{5} = 8 \]

### Step 5: Find \( b^2 \) using the eccentricity formula
The eccentricity of an ellipse is also defined as:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the known values:
\[ \frac{5}{8} = \sqrt{1 - \frac{b^2}{8^2}} \]
Squaring both sides:
\[ \left(\frac{5}{8}\right)^2 = 1 - \frac{b^2}{64} \]
This simplifies to:
\[ \frac{25}{64} = 1 - \frac{b^2}{64} \]

### Step 6: Rearranging to find \( b^2 \)
Rearranging gives:
\[ \frac{b^2}{64} = 1 - \frac{25}{64} \]
Finding a common denominator:
\[ 1 = \frac{64}{64} \implies \frac{b^2}{64} = \frac{64 - 25}{64} = \frac{39}{64} \]
Multiplying both sides by 64:
\[ b^2 = 39 \]

### Step 7: Calculate the length of the latus rectum
The length of the latus rectum \( L \) of an ellipse is given by the formula:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \( b^2 \) and \( a \):
\[ L = \frac{2 \cdot 39}{8} \]
This simplifies to:
\[ L = \frac{78}{8} = \frac{39}{4} \]

### Final Answer
Thus, the length of the latus rectum is:
\[ \boxed{\frac{39}{4}} \]