Asymptotes of a rectangular hyperbola are x = 5 and y = 4. If the hyperbola passes through (6, 8) and length of its latus rectum is I, then I is equal to

Answer

To solve the problem, we need to find the length of the latus rectum of a rectangular hyperbola given its asymptotes and a point through which it passes.

### Step-by-Step Solution:

1. **Identify the Asymptotes**:
The asymptotes of the hyperbola are given as \( x = 5 \) and \( y = 4 \). This means the hyperbola is centered at the point \( (5, 4) \).

2. **Equation of the Hyperbola**:
The standard form of a rectangular hyperbola centered at \( (h, k) \) is given by:
\[ (x - h)(y - k) = c^2 \]
Substituting \( h = 5 \) and \( k = 4 \), we have:
\[ (x - 5)(y - 4) = c^2 \]

3. **Substituting the Point (6, 8)**:
Since the hyperbola passes through the point \( (6, 8) \), we can substitute these values into the equation to find \( c^2 \):
\[ (6 - 5)(8 - 4) = c^2 \]
Simplifying this gives:
\[ (1)(4) = c^2 \implies c^2 = 4 \]

4. **Finding the Value of c**:
Taking the square root of both sides, we find:
\[ c = 2 \]

5. **Length of the Latus Rectum**:
The length of the latus rectum \( L \) of a rectangular hyperbola is given by the formula:
\[ L = 2\sqrt{2} \cdot c \]
Substituting \( c = 2 \) into the formula:
\[ L = 2\sqrt{2} \cdot 2 = 4\sqrt{2} \]

### Final Answer:
The length of the latus rectum \( L \) is \( 4\sqrt{2} \) units.

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