Two particles start from point (2, -1), one moving two units along the line x+ y = 1 and the other 5 units along the line x - 2y = 4, If the particle move towards increasing y, then their new positions are:

Explanation

To find the new positions of the two particles after they move along their respective lines, we will follow these steps:

### Step 1: Identify the initial position of the particles
Both particles start from the point (2, -1).

### Step 2: Determine the direction and distance for the first particle
The first particle moves 2 units along the line \(x + y = 1\).

1. **Find the slope of the line**:
The equation \(x + y = 1\) can be rewritten as \(y = -x + 1\). The slope (m) is -1.

2. **Calculate the angle**:
The angle \(\theta\) corresponding to this slope is given by:
\[ \tan \theta = -1 \implies \theta = 135^\circ \text{ (in the second quadrant)} \]

3. **Calculate \(\cos \theta\) and \(\sin \theta\)**:
\[ \cos \theta = -\frac{1}{\sqrt{2}}, \quad \sin \theta = \frac{1}{\sqrt{2}} \]

4. **Use parametric equations to find new coordinates**:
The new coordinates \((x_1, y_1)\) after moving 2 units are given by:
\[ x_1 = 2 + 2 \cdot \cos \theta = 2 + 2 \left(-\frac{1}{\sqrt{2}}\right) = 2 - \sqrt{2} \]
\[ y_1 = -1 + 2 \cdot \sin \theta = -1 + 2 \left(\frac{1}{\sqrt{2}}\right) = -1 + \sqrt{2} \]

### Step 3: Determine the direction and distance for the second particle
The second particle moves 5 units along the line \(x - 2y = 4\).

1. **Find the slope of the line**:
The equation \(x - 2y = 4\) can be rewritten as \(y = \frac{1}{2}x - 2\). The slope (m) is \(\frac{1}{2}\).

2. **Calculate the angle**:
The angle \(\theta\) corresponding to this slope is given by:
\[ \tan \theta = \frac{1}{2} \implies \theta = \tan^{-1}\left(\frac{1}{2}\right) \]

3. **Calculate \(\cos \theta\) and \(\sin \theta\)**:
Using the identity for \(\tan\):
\[ \cos \theta = \frac{2}{\sqrt{5}}, \quad \sin \theta = \frac{1}{\sqrt{5}} \]

4. **Use parametric equations to find new coordinates**:
The new coordinates \((x_2, y_2)\) after moving 5 units are given by:
\[ x_2 = 2 + 5 \cdot \cos \theta = 2 + 5 \left(\frac{2}{\sqrt{5}}\right) = 2 + 2\sqrt{5} \]
\[ y_2 = -1 + 5 \cdot \sin \theta = -1 + 5 \left(\frac{1}{\sqrt{5}}\right) = -1 + \sqrt{5} \]

### Final Results
The new positions of the particles are:
- First particle: \((2 - \sqrt{2}, -1 + \sqrt{2})\)
- Second particle: \((2 + 2\sqrt{5}, -1 + \sqrt{5})\)