The equation of the hyperbola is xy-4x+3y=0 and its asymptotes are xy-4x+3y=k then k= a) 3 b) -6 c)-12 d)12

Options

A. 3
B. `-6`
C. `-12`
D. 12

Answer

To solve the problem, we need to find the value of \( k \) for the asymptotes of the hyperbola given by the equation \( xy - 4x + 3y = 0 \).

### Step-by-Step Solution:

1. **Start with the given equation of the hyperbola:**
\[ xy - 4x + 3y = 0 \]

2. **Rearrange the equation:**
We can factor the equation to make it easier to analyze:
\[ xy - 4x + 3y = 0 \implies x(y - 4) + 3y = 0 \]

3. **Add and subtract 12:**
To facilitate factoring, we can add and subtract 12:
\[ x(y - 4) + 3y + 12 - 12 = 0 \implies x(y - 4) + 3(y - 4) - 12 = 0 \]

4. **Factor the expression:**
Now we can factor out \( (y - 4) \):
\[ (y - 4)(x + 3) - 12 = 0 \]

5. **Rearranging gives us the equation of the asymptote:**
\[ (y - 4)(x + 3) = 12 \]
This means that the asymptotes can be represented by the equation:
\[ xy - 4x + 3y - 12 = 0 \]

6. **Identify the value of \( k \):**
The asymptotes are given in the form \( xy - 4x + 3y = k \). From our rearranged equation, we see that:
\[ k = 12 \]

### Conclusion:
Thus, the value of \( k \) is:
\[ \boxed{12} \]