The length of latus rectum`AB` of ellipse `(x^(2))/4+(y^(2))/3=1` is :

Options

A. `3`
B. `16/5`
C. `32/5`
D. `64/5`

Answer

To find the length of the latus rectum \( AB \) of the ellipse given by the equation

\[ \frac{x^2}{4} + \frac{y^2}{3} = 1, \]

we can follow these steps:

### Step 1: Identify the standard form of the ellipse
The standard form of an ellipse is

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]

From the given equation, we can identify \( a^2 = 4 \) and \( b^2 = 3 \).

### Step 2: Determine the values of \( a \) and \( b \)
To find \( a \) and \( b \), we take the square roots of \( a^2 \) and \( b^2 \):

\[ a = \sqrt{4} = 2, \]
\[ b = \sqrt{3}. \]

### 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:

\[ L = \frac{2b^2}{a}. \]

### Step 4: Substitute the values of \( b^2 \) and \( a \)
Now, substituting the values we found:

\[ L = \frac{2 \cdot b^2}{a} = \frac{2 \cdot 3}{2}. \]

### Step 5: Simplify the expression
Simplifying the expression gives:

\[ L = \frac{6}{2} = 3. \]

### Conclusion
Thus, the length of the latus rectum \( AB \) of the ellipse is

\[ \boxed{3}. \]