Write the length o the latus rectum of the hyperbola `16 x^2-9y^2=144.`

Explanation

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

### Step 1: Rewrite the equation in standard form
We start with the equation of the hyperbola:
\[ 16x^2 - 9y^2 = 144 \]
To rewrite this in standard form, we divide the entire equation by 144:
\[ \frac{16x^2}{144} - \frac{9y^2}{144} = 1 \]
This simplifies to:
\[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we can identify:
\[ a^2 = 9 \quad \text{and} \quad b^2 = 16 \]
Thus, we find:
\[ a = \sqrt{9} = 3 \quad \text{and} \quad b = \sqrt{16} = 4 \]

### Step 3: Use the formula for the length of the latus rectum
The formula for the length of the latus rectum \(L\) of a hyperbola is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \(b^2\) and \(a\):
\[ L = \frac{2 \times 16}{3} \]
Calculating this gives:
\[ L = \frac{32}{3} \]

### Conclusion
The length of the latus rectum of the given hyperbola is:
\[ \frac{32}{3} \]

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