The length of the latusrectum `16x^(2)+25y^(2) = 400` is

Options

A. `25/2`
B. `25/4`
C. `16/5`
D. `32/5`

Answer

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

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

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 25\) (since it is under \(x^2\))
- \(b^2 = 16\) (since it is under \(y^2\))

### Step 3: Determine the values of \(a\) and \(b\)
Next, we find \(a\) and \(b\):
\[ a = \sqrt{25} = 5 \]
\[ b = \sqrt{16} = 4 \]

### Step 4: Use the formula for the length of the latus rectum
The formula for 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 \(a\):
\[ L = \frac{2 \times 16}{5} \]

### Step 5: Calculate the length of the latus rectum
Now we calculate:
\[ L = \frac{32}{5} \]

### Final Answer
Thus, the length of the latus rectum is:
\[ \frac{32}{5} \]