Find the length of the latus rectum of the ellipse `x^2/36+y^2/25=1`

Answer

To find the length of the latus rectum of the ellipse given by the equation \( \frac{x^2}{36} + \frac{y^2}{25} = 1 \), we can follow these steps:

### Step 1: Identify the values of \( a^2 \) and \( b^2 \)
The standard form of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
From the equation \( \frac{x^2}{36} + \frac{y^2}{25} = 1 \), we can identify:
- \( a^2 = 36 \)
- \( b^2 = 25 \)

### Step 2: Calculate \( a \) and \( b \)
Now, we can find the values of \( a \) and \( b \):
\[ a = \sqrt{36} = 6 \]
\[ b = \sqrt{25} = 5 \]

### Step 3: Determine if \( a > b \)
Since \( a = 6 \) and \( b = 5 \), we have \( a > b \). This means we will use the formula for the length of the latus rectum for ellipses where \( a > b \).

### Step 4: Use the formula for the length of the latus rectum
The length of the latus rectum \( L \) for an ellipse where \( a > b \) is given by the formula:
\[ L = \frac{2b^2}{a} \]

### Step 5: Substitute the values into the formula
Now, substituting the values of \( b^2 \) and \( a \):
\[ L = \frac{2 \times 25}{6} \]
\[ L = \frac{50}{6} = \frac{25}{3} \]

### Final Answer
Thus, the length of the latus rectum of the given ellipse is:
\[ \frac{25}{3} \text{ units} \]

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