The equation of two sides of a square whose area is 25 sq. units are `3x-4y=0` an d `4x+3y=0`. Then equations of other two sides of the square are

Options

A. `3x-4y+-25=0, 4x+3y+-25=0`
B. `3x-4y+-5=0, 4x+3y+-5=0`
C. `3x-4y+-5=0, 4x+3y+-25=0`
D. none

Answer

To find the equations of the other two sides of the square given the equations of two sides, we can follow these steps:

### Step 1: Identify the given equations
The equations of the two sides of the square are:
1. \( 3x - 4y = 0 \) (let's call this line A)
2. \( 4x + 3y = 0 \) (let's call this line B)

### Step 2: Determine the slope of the given lines
- For line A: \( 3x - 4y = 0 \) can be rewritten as \( y = \frac{3}{4}x \). The slope (m1) is \( \frac{3}{4} \).
- For line B: \( 4x + 3y = 0 \) can be rewritten as \( y = -\frac{4}{3}x \). The slope (m2) is \( -\frac{4}{3} \).

### Step 3: Find the slopes of the other two sides of the square
Since the sides of a square are perpendicular to each other, the slopes of the other two sides will be the negative reciprocals of the slopes of the given lines.
- The slope of the line parallel to line A (let's call it line C) will be \( -\frac{4}{3} \).
- The slope of the line parallel to line B (let's call it line D) will be \( \frac{3}{4} \).

### Step 4: Write the equations of the new lines
We need to find the equations of lines C and D, which are parallel to lines A and B, respectively.

1. **Equation of line C** (parallel to line A):
- The general form of the equation is \( 3x - 4y + k = 0 \).
- Since the area of the square is 25 square units, the length of each side (let's denote it as \( \lambda \)) is \( \sqrt{25} = 5 \). Thus, \( k \) can be \( \pm 5 \).
- Therefore, the equations of line C are:
- \( 3x - 4y + 5 = 0 \)
- \( 3x - 4y - 5 = 0 \)

2. **Equation of line D** (parallel to line B):
- The general form of the equation is \( 4x + 3y + m = 0 \).
- Similarly, \( m \) can also be \( \pm 5 \).
- Therefore, the equations of line D are:
- \( 4x + 3y + 5 = 0 \)
- \( 4x + 3y - 5 = 0 \)

### Step 5: Final equations of the other two sides
Combining the results, the equations of the other two sides of the square are:
1. \( 3x - 4y + 5 = 0 \)
2. \( 3x - 4y - 5 = 0 \)
3. \( 4x + 3y + 5 = 0 \)
4. \( 4x + 3y - 5 = 0 \)