Find the length of the latus rectum of the ellipse `9x^(2)+16y^(2)=144`

Answer

To find the length of the latus rectum of the ellipse given by the equation \(9x^2 + 16y^2 = 144\), we can follow these steps:

### Step 1: Write the equation of the ellipse
The given equation is:
\[ 9x^2 + 16y^2 = 144 \]

### Step 2: Convert to standard form
To convert this equation into the standard form of an ellipse, we divide the entire equation by 144:
\[ \frac{9x^2}{144} + \frac{16y^2}{144} = \frac{144}{144} \]
This simplifies to:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

### Step 3: Identify \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 16\) which gives \(a = 4\)
- \(b^2 = 9\) which gives \(b = 3\)

### Step 4: 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:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \(b^2\) and \(a\):
\[ L = \frac{2 \times 9}{4} \]

### Step 5: Simplify the expression
Calculating the above expression:
\[ L = \frac{18}{4} = \frac{9}{2} \]

### Final Answer
Thus, the length of the latus rectum of the ellipse is:
\[ \frac{9}{2} \]