What is the length of the latus rectum of an ellipse `25x^(2)+16y^(2)=400`?

Options

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

Answer

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

### Step 1: Rewrite the equation in standard form
We start with the equation of the ellipse:
\[ 25x^2 + 16y^2 = 400 \]
To convert it into standard form, we divide the entire equation 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 \(a^2\) and \(b^2\)
From the standard form of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 16\) which gives \(a = 4\)
- \(b^2 = 25\) which gives \(b = 5\)

### Step 3: Determine the length of the latus rectum
For an ellipse, the length of the latus rectum \(L\) is given by the formula:
\[ L = \frac{2a^2}{b} \]
Substituting the values of \(a\) and \(b\):
\[ L = \frac{2 \times 16}{5} = \frac{32}{5} \]

### Step 4: Conclusion
Thus, the length of the latus rectum of the ellipse is:
\[ \frac{32}{5} \]