Find the asymptotes and axes of hyperbola having equation `xy-3y-4x+7=0`.

Explanation

To find the asymptotes and axes of the hyperbola given by the equation \( xy - 3y - 4x + 7 = 0 \), we can follow these steps:

### Step 1: Rearrange the equation
Start with the given equation:
\[ xy - 3y - 4x + 7 = 0 \]
Rearranging gives:
\[ xy - 3y = 4x - 7 \]

### Step 2: Factor the equation
Factor out \(y\) from the left side:
\[ y(x - 3) = 4x - 7 \]
Now, we can isolate \(y\):
\[ y = \frac{4x - 7}{x - 3} \]

### Step 3: Identify the type of hyperbola
This equation can be rewritten in the form of a rectangular hyperbola:
\[ xy = 4x - 7 + 3y \]
Rearranging gives:
\[ xy - 3y - 4x + 7 = 0 \]
This indicates that it is a rectangular hyperbola.

### Step 4: Find the asymptotes
For a rectangular hyperbola of the form \(XY = C\), the asymptotes are given by setting the factors equal to zero. Here, we can find the asymptotes by solving:
1. Set \(x - 3 = 0\):
\[ x = 3 \]
2. Set \(y - 4 = 0\):
\[ y = 4 \]

Thus, the asymptotes are:
\[ x = 3 \quad \text{and} \quad y = 4 \]

### Step 5: Find the center of the hyperbola
The center of the hyperbola is at the point where the asymptotes intersect:
\[ (3, 4) \]

### Step 6: Determine the axes of the hyperbola
Since this is a rectangular hyperbola, the axes bisect the asymptotes. The slopes of the axes will be \(+1\) and \(-1\).

1. For the slope \(+1\):
\[ y - 4 = 1(x - 3) \implies y = x + 1 \]
Rearranging gives:
\[ x - y + 1 = 0 \]

2. For the slope \(-1\):
\[ y - 4 = -1(x - 3) \implies y = -x + 7 \]
Rearranging gives:
\[ x + y - 7 = 0 \]

### Final Result
The asymptotes of the hyperbola are:
\[ x = 3 \quad \text{and} \quad y = 4 \]
The axes of the hyperbola are:
\[ x - y + 1 = 0 \quad \text{and} \quad x + y - 7 = 0 \]