If `3x+4y+3=0,3x+4y-7=0` and `4x-3y-2=0` be the three sides of a square, then the equation of the fourth side is

Options

A. `4x-3y-12=0`
B. `4x-3y+8=0`
C. `4x-3y-10=0`
D. `4x-3y+6=0`

Answer

To find the equation of the fourth side of the square given the equations of the three sides, we can follow these steps:

### Step 1: Identify the given lines
The equations of the three sides of the square are:
1. \( 3x + 4y + 3 = 0 \) (Line 1)
2. \( 3x + 4y - 7 = 0 \) (Line 2)
3. \( 4x - 3y - 2 = 0 \) (Line 3)

### Step 2: Find the distance between the parallel lines
The first two lines are parallel since they have the same coefficients for \(x\) and \(y\). The distance \(d\) between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by the formula:

\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

For our lines:
- \(C_1 = 3\) (from Line 1)
- \(C_2 = -7\) (from Line 2)

Calculating the distance:

\[ d = \frac{|-7 - 3|}{\sqrt{3^2 + 4^2}} = \frac{|-10|}{\sqrt{9 + 16}} = \frac{10}{5} = 2 \]

### Step 3: Determine the length of the side of the square
Since the distance between the two parallel lines is 2, this represents the length of one side of the square.

### Step 4: Find the fourth side of the square
The third line \(4x - 3y - 2 = 0\) is not parallel to the first two lines. We need to find the equation of the fourth side of the square, which will also be parallel to the line \(4x - 3y - 2 = 0\).

The general form of the equation of a line parallel to \(4x - 3y - 2 = 0\) is:

\[ 4x - 3y + C = 0 \]

### Step 5: Calculate the distance from the third line to the fourth side
The distance from the line \(4x - 3y - 2 = 0\) to the new line \(4x - 3y + C = 0\) must equal the side length of the square, which is 2. Using the distance formula again:

\[ \frac{|C - (-2)|}{\sqrt{4^2 + (-3)^2}} = 2 \]

This simplifies to:

\[ \frac{|C + 2|}{5} = 2 \]

### Step 6: Solve for \(C\)
Multiplying both sides by 5 gives:

\[ |C + 2| = 10 \]

This results in two cases:

1. \(C + 2 = 10 \Rightarrow C = 8\)
2. \(C + 2 = -10 \Rightarrow C = -12\)

### Step 7: Write the equations of the fourth side
Thus, we have two possible equations for the fourth side:

1. \(4x - 3y + 8 = 0\)
2. \(4x - 3y - 12 = 0\)

### Conclusion
The equation of the fourth side of the square can be either:

1. \(4x - 3y + 8 = 0\)
2. \(4x - 3y - 12 = 0\)