The length of the latus rectum of the hyperbola `25x^(2)-16y^(2)=400` is

Options

A. `8//3`
B. `9//2`
C. `11//3`
D. `25//2`

Answer

To find the length of the latus rectum of the hyperbola given by the equation \( 25x^2 - 16y^2 = 400 \), we will follow these steps:

### Step 1: Write the equation in standard form
We start with the equation:
\[ 25x^2 - 16y^2 = 400 \]
To convert this into standard form, we divide every term by 400:
\[ \frac{25x^2}{400} - \frac{16y^2}{400} = 1 \]
This simplifies to:
\[ \frac{x^2}{16} - \frac{y^2}{25} = 1 \]

### Step 2: Identify the parameters \( a \) and \( b \)
From the standard form of the hyperbola:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
we can identify:
\[ a^2 = 16 \quad \Rightarrow \quad a = 4 \]
\[ b^2 = 25 \quad \Rightarrow \quad b = 5 \]

### Step 3: Use the formula for the length of the latus rectum
The length of the latus rectum \( L \) of a hyperbola is given by the formula:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \( b \) and \( a \):
\[ L = \frac{2 \times 25}{4} \]

### Step 4: Calculate the length of the latus rectum
Now we perform the calculation:
\[ L = \frac{50}{4} = 12.5 \]

### Final Answer
Thus, the length of the latus rectum of the hyperbola is:
\[ \boxed{12.5} \]