The graph of `xy-4x-2y-4=0` can be expressed as a set of parametic. If `y=(4t)/(t-3)` and `x=f(t),` then f(t)=

Options

A. `t+1`
B. `t-1`
C. `3t-3`
D. `(t-3)/(4t)`

Explanation

To solve the problem, we need to express the equation \( xy - 4x - 2y - 4 = 0 \) in terms of the parameter \( t \) where \( y = \frac{4t}{t-3} \) and find \( x = f(t) \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ xy - 4x - 2y - 4 = 0 \]

2. **Substitute \( y \) with \( \frac{4t}{t-3} \):**
\[ x \left(\frac{4t}{t-3}\right) - 4x - 2\left(\frac{4t}{t-3}\right) - 4 = 0 \]

3. **Multiply through by \( t - 3 \) to eliminate the fraction:**
\[ 4tx - 4x(t - 3) - 8t - 4(t - 3) = 0 \]

4. **Distribute and simplify:**
\[ 4tx - 4tx + 12x - 8t - 4t + 12 = 0 \]
This simplifies to:
\[ 12x - 12t + 12 = 0 \]

5. **Rearranging the equation:**
\[ 12x = 12t - 12 \]

6. **Divide by 12:**
\[ x = t - 1 \]

7. **Thus, we find:**
\[ f(t) = t - 1 \]

### Final Answer:
\[ f(t) = t - 1 \]