How are the points (3,4) and (2,-6) situated w.r.t the line `3x-4y-8=0`?

Options

A. same side
B. opposite side
C. one on the line
D. none

Answer

To determine how the points (3, 4) and (2, -6) are situated with respect to the line given by the equation \(3x - 4y - 8 = 0\), we will evaluate the positions of these points relative to the line.

### Step 1: Identify the line equation
The line is given by the equation:
\[ 3x - 4y - 8 = 0 \]

### Step 2: Substitute the coordinates of point P (3, 4) into the line equation
We substitute \(x = 3\) and \(y = 4\) into the equation:
\[ 3(3) - 4(4) - 8 = 9 - 16 - 8 = -15 \]
Since \(-15 < 0\), point P (3, 4) lies on one side of the line.

### Step 3: Substitute the coordinates of point Q (2, -6) into the line equation
Next, we substitute \(x = 2\) and \(y = -6\) into the equation:
\[ 3(2) - 4(-6) - 8 = 6 + 24 - 8 = 22 \]
Since \(22 > 0\), point Q (2, -6) lies on the opposite side of the line compared to point P.

### Step 4: Conclusion
Since point P (3, 4) is on one side of the line and point Q (2, -6) is on the opposite side, we conclude that:
- The points (3, 4) and (2, -6) are situated on opposite sides of the line \(3x - 4y - 8 = 0\).

### Final Answer
Both points lie on opposite sides of the line.

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