Equation of conjugate axis of hypobola `xy -3y-4x+7=0` is

Options

A. `y + x=3`
B. `y + x =7`
C. `y-x=3`
D. none of these

Answer

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

### Step 1: Rewrite the equation in standard form
We start with the given equation:
\[ xy - 3y - 4x + 7 = 0 \]
Rearranging this, we can express \( y \) in terms of \( x \):
\[ xy - 3y = 4x - 7 \]
Factoring out \( y \) from the left side:
\[ y(x - 3) = 4x - 7 \]
Thus,
\[ y = \frac{4x - 7}{x - 3} \]

### Step 2: Identify the center of the hyperbola
To find the center, we can analyze the asymptotes. The hyperbola can be expressed in the form of \( (x - h)(y - k) = c \) where \( (h, k) \) is the center.

From the rearranged equation, we can see that the hyperbola has asymptotes at:
\[ x - 3 = 0 \quad \text{and} \quad y - 4 = 0 \]
Thus, the center of the hyperbola is at the point \( (3, 4) \).

### Step 3: Determine the slopes of the asymptotes
For a rectangular hyperbola, the slopes of the asymptotes are \( \pm 1 \). Therefore, the equations of the asymptotes can be written as:
\[ y - 4 = 1(x - 3) \quad \text{and} \quad y - 4 = -1(x - 3) \]
This gives us the two asymptotes:
1. \( y = x + 1 \)
2. \( y = -x + 7 \)

### Step 4: Find the conjugate axis
In a rectangular hyperbola, the conjugate axis is perpendicular to the transverse axis and passes through the center. The equation of the conjugate axis can be derived from the center and the slopes of the asymptotes.

Since the slopes of the asymptotes are \( \pm 1 \), the conjugate axis will have a slope of \( 0 \) (horizontal line). Therefore, the equation of the conjugate axis is:
\[ y = k \]
where \( k \) is the y-coordinate of the center. Since the center is \( (3, 4) \), we have:
\[ y = 4 \]

### Final Answer
The equation of the conjugate axis of the hyperbola \( xy - 3y - 4x + 7 = 0 \) is:
\[ y = 4 \]