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

Options

A. `3x-4ypm25=0, 4x+3y pm 25=0`
B. `3x-4ypm05=0, 4x+3y pm 5=0`
C. `3x-4y pm 05=0 , 4x+3y pm=0`
D. none of these

Answer

To find the equations of the other two sides of a 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. \( 3x - 4y = 0 \)
2. \( 4x + 3y = 0 \)

### Step 2: Determine the Slopes of the Given Lines
We can rewrite the equations in slope-intercept form \( y = mx + b \) to find their slopes.

1. For \( 3x - 4y = 0 \):
\[ 4y = 3x \implies y = \frac{3}{4}x \]
The slope \( m_1 = \frac{3}{4} \).

2. For \( 4x + 3y = 0 \):
\[ 3y = -4x \implies y = -\frac{4}{3}x \]
The slope \( m_2 = -\frac{4}{3} \).

### Step 3: Find the Length of the Side of the Square
The area of the square is given as \( 25 \) sq. units. The side length \( a \) can be calculated as:
\[ a^2 = 25 \implies a = 5 \]

### Step 4: Calculate the Distance Between the Parallel Lines
The distance \( d \) between two parallel lines can be calculated using the formula:
\[ d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \]
For the first line \( 3x - 4y = 0 \) (rewritten as \( 3x - 4y + 0 = 0 \)), \( c_1 = 0 \).
For the second line \( 4x + 3y = 0 \) (rewritten as \( 4x + 3y + 0 = 0 \)), \( c_2 = 0 \).

### Step 5: Set Up the Equations for the Other Two Sides
The equations of the other two sides will be parallel to the given lines. Since the distance between the sides of the square is equal to the side length \( a = 5 \), we can write:

1. For the line parallel to \( 3x - 4y = 0 \):
\[ 3x - 4y + d = 0 \quad \text{where } d = \pm 25 \]
Thus, the equations become:
\[ 3x - 4y + 25 = 0 \quad \text{and} \quad 3x - 4y - 25 = 0 \]

2. For the line parallel to \( 4x + 3y = 0 \):
\[ 4x + 3y + d = 0 \quad \text{where } d = \pm 25 \]
Thus, the equations become:
\[ 4x + 3y + 25 = 0 \quad \text{and} \quad 4x + 3y - 25 = 0 \]

### Final Equations
The equations of the other two sides of the square are:
1. \( 3x - 4y + 25 = 0 \)
2. \( 3x - 4y - 25 = 0 \)
3. \( 4x + 3y + 25 = 0 \)
4. \( 4x + 3y - 25 = 0 \)