Find the length of the latus -rectum of the ellipse :
`(x^(2))/(4) + (y^(2))/(9)` = 1 .

Answer

To find the length of the latus rectum of the ellipse given by the equation

\[ \frac{x^2}{4} + \frac{y^2}{9} = 1, \]

we will follow these steps:

### Step 1: Identify the values of \(a\) and \(b\)

The standard form of the ellipse is

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

From the given equation, we can identify \(a^2 = 4\) and \(b^2 = 9\). Therefore, we find:

\[ a = \sqrt{4} = 2, \]
\[ b = \sqrt{9} = 3. \]

### Step 2: Determine the orientation of the ellipse

Since \(b > a\) (3 > 2), the major axis is along the y-axis.

### 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:

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

### Step 4: Substitute the values of \(a\) and \(b\)

Now, we substitute \(a\) and \(b\) into the formula:

\[ L = \frac{2 \cdot (2^2)}{3} = \frac{2 \cdot 4}{3} = \frac{8}{3}. \]

### Step 5: Conclusion

Thus, the length of the latus rectum of the given ellipse is

\[ \frac{8}{3}. \]

---