The four sides of a quadrilateral are given by the equation `xy(x-2)(y - 3)=0` The equation of the line parallel to `x-4y=0` that divides the quadrilateral in two equal areas is

Options

A. `x-4y-1=0`
B. `x-4y+5=0`
C. `x-4y+1=0`
D. `x-4y+3=0`

Answer

To solve the problem, we need to find the equation of a line parallel to \( x - 4y = 0 \) that divides the quadrilateral defined by the equation \( xy(x-2)(y-3) = 0 \) into two equal areas.

### Step-by-Step Solution:

1. **Identify the Quadrilateral**:
The equation \( xy(x-2)(y-3) = 0 \) gives us the lines:
- \( x = 0 \) (the y-axis)
- \( y = 0 \) (the x-axis)
- \( x = 2 \) (a vertical line)
- \( y = 3 \) (a horizontal line)

These lines form a rectangle with vertices at:
- \( A(0, 0) \)
- \( B(2, 0) \)
- \( C(2, 3) \)
- \( D(0, 3) \)

2. **Calculate the Area of the Quadrilateral**:
The area \( A \) of the rectangle can be calculated as:
\[ A = \text{length} \times \text{width} = 2 \times 3 = 6 \]

3. **Determine the Area to be Divided**:
Since we want to divide the quadrilateral into two equal areas, each area should be:
\[ \frac{6}{2} = 3 \]

4. **Identify the Slope of the Required Line**:
The line \( x - 4y = 0 \) can be rewritten as \( y = \frac{1}{4}x \). The slope \( m \) of this line is \( \frac{1}{4} \). The line we are looking for must also have this slope.

5. **Set Up the Equation of the Line**:
The line can be expressed in point-slope form as:
\[ y - h = \frac{1}{4}(x - 0) \quad \text{(since it passes through the y-axis)} \]
This simplifies to:
\[ y = \frac{1}{4}x + h \]

6. **Find the Intersection Points**:
The line intersects the sides of the rectangle. We need to find the intersection with \( x = 2 \):
\[ y = \frac{1}{4}(2) + h = \frac{1}{2} + h \]
The intersection point is \( (2, \frac{1}{2} + h) \).

7. **Calculate the Area of Triangle \( A, B, P, Q \)**:
The area of triangle \( A(0, 0), B(2, 0), P(2, \frac{1}{2} + h) \) can be calculated using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \( 2 \) and the height is \( \frac{1}{2} + h \):
\[ \text{Area} = \frac{1}{2} \times 2 \times \left(\frac{1}{2} + h\right) = \frac{1}{2} + h \]

8. **Set Up the Equation for Equal Areas**:
We want this area to equal \( 3 \):
\[ \frac{1}{2} + h = 3 \]
Solving for \( h \):
\[ h = 3 - \frac{1}{2} = \frac{5}{2} \]

9. **Substitute \( h \) Back into the Line Equation**:
The equation of the line becomes:
\[ y = \frac{1}{4}x + \frac{5}{2} \]

10. **Convert to Standard Form**:
Rearranging gives:
\[ 4y - x + 10 = 0 \]
or
\[ x - 4y + 10 = 0 \]

### Final Answer:
The equation of the line parallel to \( x - 4y = 0 \) that divides the quadrilateral into two equal areas is:
\[ x - 4y + 10 = 0 \]