If the sides of a square lie along the lines `5x-12y-65=0` and `5x-12y+26=0` then its area is

Options

A. `3^(2)`
B. `4^(2)`
C. `7^(2)`
D. `9^(2)`

Answer

To find the area of the square whose sides lie along the given parallel lines, we will follow these steps:

### Step 1: Identify the equations of the lines
The equations of the lines are:
1. \(5x - 12y - 65 = 0\) (Line 1)
2. \(5x - 12y + 26 = 0\) (Line 2)

### Step 2: Confirm that the lines are parallel
The coefficients of \(x\) and \(y\) in both equations are the same, which confirms that the lines are parallel.

### Step 3: Calculate the distance between the two parallel lines
The formula for the distance \(D\) between two parallel lines given by \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is:
\[ D = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

Here, \(A = 5\), \(B = -12\), \(C_1 = -65\), and \(C_2 = 26\).

### Step 4: Substitute the values into the distance formula
Substituting the values into the formula:
\[ D = \frac{|26 - (-65)|}{\sqrt{5^2 + (-12)^2}} = \frac{|26 + 65|}{\sqrt{25 + 144}} = \frac{91}{\sqrt{169}} \]

### Step 5: Simplify the distance
Calculating the square root:
\[ \sqrt{169} = 13 \]
Thus, the distance becomes:
\[ D = \frac{91}{13} = 7 \]

### Step 6: Calculate the area of the square
Since the distance between the two lines is the side length of the square, the area \(A\) of the square is given by:
\[ A = \text{side}^2 = D^2 = 7^2 = 49 \]

### Final Answer
The area of the square is \(49\) square units.

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