Three vertices of a quadrilateral in order are `(6,1)(7,2)" and "(-1,0)`. If the area of the quadrilateral is 4 sq. unit. Then the locus of the fourth vertex has the equation.

Options

A. x-7 y=1
B. x-7 y+15=0
C. x=7y+15=0
D. `(x-7y)^(2)+14(x-7y)-15=0`

Answer

To solve the problem, we need to find the locus of the fourth vertex of a quadrilateral given three vertices and the area of the quadrilateral. The vertices provided are \( A(6, 1) \), \( B(7, 2) \), and \( C(-1, 0) \), and the area is given as 4 square units.

### Step-by-Step Solution:

1. **Identify the Vertices**:
We have three vertices of the quadrilateral:
- \( A(6, 1) \)
- \( B(7, 2) \)
- \( C(-1, 0) \)
Let the fourth vertex be \( D(h, k) \).

2. **Use the Area Formula**:
The area \( A \) of a quadrilateral with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| \]
Substituting the coordinates:
\[ A = \frac{1}{2} \left| 6 \cdot 2 + 7 \cdot 0 + (-1) \cdot k + h \cdot 1 - (1 \cdot 7 + 2 \cdot (-1) + 0 \cdot h + k \cdot 6) \right| \]

3. **Simplify the Area Expression**:
Expanding the expression:
\[ A = \frac{1}{2} \left| 12 + 0 - k + h - (7 - 2 + 0 + 6k) \right| \]
This simplifies to:
\[ A = \frac{1}{2} \left| h - k + 12 - 7 + 2 - 6k \right| \]
Which further simplifies to:
\[ A = \frac{1}{2} \left| h - 7k + 7 \right| \]

4. **Set the Area Equal to 4**:
Given that the area is 4 square units:
\[ \frac{1}{2} \left| h - 7k + 7 \right| = 4 \]
Multiplying both sides by 2 gives:
\[ \left| h - 7k + 7 \right| = 8 \]

5. **Consider the Two Cases for Absolute Value**:
This leads to two equations:
- Case 1: \( h - 7k + 7 = 8 \)
- Case 2: \( h - 7k + 7 = -8 \)

**From Case 1**:
\[ h - 7k = 1 \quad \text{(Equation 1)} \]

**From Case 2**:
\[ h - 7k = -15 \quad \text{(Equation 2)} \]

6. **Rewrite the Equations**:
- Equation 1 can be rewritten as:
\[ h = 7k + 1 \]
- Equation 2 can be rewritten as:
\[ h = 7k - 15 \]

### Final Locus Equations:
The locus of the fourth vertex \( D(h, k) \) is given by the two equations:
1. \( h - 7k = 1 \)
2. \( h - 7k = -15 \)