Find the latus rectum of ellipse
`(x^(2))/(36) + (y^(2))/(16) = 1`.

Answer

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

\[ \frac{x^2}{36} + \frac{y^2}{16} = 1, \]

we can follow these steps:

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

The standard form of the ellipse is

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

where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

From the given equation, we can see that:

- \(a^2 = 36\) which means \(a = \sqrt{36} = 6\)
- \(b^2 = 16\) which means \(b = \sqrt{16} = 4\)

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

Since \(a > b\) (6 > 4), this indicates that the ellipse is oriented horizontally.

### Step 3: Use the formula for the length of the latus rectum

The length of the latus rectum \(L\) of an ellipse is given by the formula:

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

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

Now we can substitute the values we found:

\[ L = \frac{2b^2}{a} = \frac{2 \times (4^2)}{6}. \]

Calculating \(b^2\):

\[ b^2 = 4^2 = 16. \]

Now substituting this into the formula:

\[ L = \frac{2 \times 16}{6} = \frac{32}{6} = \frac{16}{3}. \]

### Final Answer

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

\[ \frac{16}{3}. \]

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