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

Options

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

Answer

To determine the 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: Rewrite the line equation in standard form
The equation of the line can be rewritten as:
\[ 4x + 2y - 1 = 0 \]
Here, \(a = 4\), \(b = 2\), and \(c = -1\).

### Step 2: Substitute the first point (1, 2) into the line equation
Let’s substitute \(P_1(1, 2)\) into the equation:
\[ 4(1) + 2(2) - 1 \]
Calculating this gives:
\[ 4 + 4 - 1 = 7 \]
Since \(7 > 0\), point \(P_1\) is on one side of the line.

### Step 3: Substitute the second point (2, 1) into the line equation
Now, let’s substitute \(P_2(2, 1)\) into the equation:
\[ 4(2) + 2(1) - 1 \]
Calculating this gives:
\[ 8 + 2 - 1 = 9 \]
Since \(9 > 0\), point \(P_2\) is also on the same side of the line.

### Step 4: Conclusion
Both points \(P_1(1, 2)\) and \(P_2(2, 1)\) yield positive results when substituted into the line equation. Therefore, both points lie on the same side of the line \(4x + 2y = 1\).

### Summary of Results
- Point \(P_1(1, 2)\): \(7 > 0\) (same side)
- Point \(P_2(2, 1)\): \(9 > 0\) (same side)

Thus, we conclude that both points are on the same side of the line.

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