Points (1,2) and (-2,1) are

Options

A. on the same side of the `4x+2y=1`
B. on the line `4x+2y=1`
C. on the opposite side of `4x+2y=1`
D. none of these

Explanation

To determine the relative position of the points (1, 2) and (-2, 1) with respect to the line given by the equation \(4x + 2y = 1\), we will follow these steps:

### Step 1: Write the equation of the line in standard form
The equation of the line is already given as:
\[ 4x + 2y - 1 = 0 \]

### Step 2: Substitute the first point (1, 2) into the line equation
We will substitute \(x = 1\) and \(y = 2\) into the line equation:
\[ L_1 = 4(1) + 2(2) - 1 \]
Calculating this gives:
\[ L_1 = 4 + 4 - 1 = 7 \]

### Step 3: Determine the sign of \(L_1\)
Since \(L_1 = 7\) which is greater than 0, the point (1, 2) is above the line.

### Step 4: Substitute the second point (-2, 1) into the line equation
Now, we will substitute \(x = -2\) and \(y = 1\) into the line equation:
\[ L_2 = 4(-2) + 2(1) - 1 \]
Calculating this gives:
\[ L_2 = -8 + 2 - 1 = -7 \]

### Step 5: Determine the sign of \(L_2\)
Since \(L_2 = -7\) which is less than 0, the point (-2, 1) is below the line.

### Step 6: Compare the signs of \(L_1\) and \(L_2\)
We have:
- \(L_1 > 0\) (point (1, 2) is above the line)
- \(L_2 < 0\) (point (-2, 1) is below the line)

Since the signs of \(L_1\) and \(L_2\) are opposite, we conclude that the points (1, 2) and (-2, 1) are on opposite sides of the line.

### Final Conclusion
The points (1, 2) and (-2, 1) are on opposite sides of the line represented by the equation \(4x + 2y = 1\).

---