The latus -rectum of the hyperbola `16x^(2) - 9y^(2) = 576 ` is `(64)/(5) `

Answer

To find the latus rectum of the hyperbola given by the equation \( 16x^2 - 9y^2 = 576 \), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the hyperbola:
\[ 16x^2 - 9y^2 = 576 \]
To convert this into standard form, we divide every term by 576:
\[ \frac{16x^2}{576} - \frac{9y^2}{576} = 1 \]
This simplifies to:
\[ \frac{x^2}{36} - \frac{y^2}{64} = 1 \]

### 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:
- \( a^2 = 36 \) which gives \( a = 6 \)
- \( b^2 = 64 \) which gives \( b = 8 \)

### Step 3: Use the formula for 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 \( a \):
\[ L = \frac{2 \times 64}{6} \]

### Step 4: Calculate the latus rectum
Now we calculate:
\[ L = \frac{128}{6} = \frac{64}{3} \]

### Conclusion
Thus, the length of the latus rectum of the hyperbola \( 16x^2 - 9y^2 = 576 \) is:
\[ \frac{64}{3} \]