Statement I The points (3,2) and (1,4) lie on opposite side of the line `3x-2y-1 =0`
Statement II The algebraic perpendicular distance from the given the point to the line have opposite sign

Options

A. Statement I is true ,statement II is true , statement II is a correct explanation for statement I
B. Statement I is true ,statement II is true statement II is not a correct explanation for statement I
C. Statement I is true ,statement II is false
D. Statement I is false ,statement II is true

Answer

To determine whether the points (3,2) and (1,4) lie on opposite sides of the line given by the equation \(3x - 2y - 1 = 0\), we can use the concept of the algebraic distance from a point to a line. Here’s a step-by-step solution:

### Step 1: Identify the line equation and points
The line equation is given as:
\[ 3x - 2y - 1 = 0 \]
The points we need to check are:
- Point A: \( (3, 2) \)
- Point B: \( (1, 4) \)

### Step 2: Calculate the algebraic distance for Point A (3, 2)
To find the algebraic distance of point \( (3, 2) \) from the line, we substitute \( x = 3 \) and \( y = 2 \) into the line equation:
\[ D_A = 3(3) - 2(2) - 1 \]
Calculating this gives:
\[ D_A = 9 - 4 - 1 = 4 \]
Since \( D_A > 0 \), point \( (3, 2) \) lies above the line.

### Step 3: Calculate the algebraic distance for Point B (1, 4)
Now, we substitute \( x = 1 \) and \( y = 4 \) into the line equation:
\[ D_B = 3(1) - 2(4) - 1 \]
Calculating this gives:
\[ D_B = 3 - 8 - 1 = -6 \]
Since \( D_B < 0 \), point \( (1, 4) \) lies below the line.

### Step 4: Conclusion
Since point \( (3, 2) \) has a positive distance from the line and point \( (1, 4) \) has a negative distance, it confirms that the two points lie on opposite sides of the line \( 3x - 2y - 1 = 0 \).

### Final Answer
Both statements are true:
- Statement I: The points (3,2) and (1,4) lie on opposite sides of the line \( 3x - 2y - 1 = 0 \).
- Statement II: The algebraic perpendicular distance from the given points to the line have opposite signs.