The length of the latus rectum of `16x^(2) + y^(2) = 16` is

Options

A. `(1)/(2)`
B. 32
C. 8
D. 16

Answer

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

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

### Step 2: Identify the values of \(a\) and \(b\)
From the standard form of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we can identify:
- \(a^2 = 1\) which gives \(a = 1\)
- \(b^2 = 16\) which gives \(b = 4\)

### Step 3: 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:
\[ L = \frac{2a^2}{b} \]
Substituting the values of \(a\) and \(b\):
\[ L = \frac{2 \cdot (1)^2}{4} \]
This simplifies to:
\[ L = \frac{2 \cdot 1}{4} = \frac{2}{4} = \frac{1}{2} \]

### Conclusion
Thus, the length of the latus rectum of the ellipse \(16x^2 + y^2 = 16\) is \(\frac{1}{2}\).

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