Find the equation of the line for which
`(i)p=3 and alpha=45^(@)` (ii)`p=5 and alpha=135^(@)`
(iii)`p=8 alpha=150^(@)` (iv) `p=3 and alpha=225^(@)`
(v)`p=2 and alpha=300^(@)` (vi)`p=4 and alpha=180^(@)`

Answer

To find the equations of the lines given the parameters \( p \) and \( \alpha \), we will use the formula:

\[ x \cos \alpha + y \sin \alpha = p \]

We will calculate the equations step by step for each case.

### (i) For \( p = 3 \) and \( \alpha = 45^\circ \)

1. **Calculate \( \cos 45^\circ \) and \( \sin 45^\circ \)**:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}}, \quad \sin 45^\circ = \frac{1}{\sqrt{2}} \]

2. **Substitute into the equation**:
\[ x \cdot \frac{1}{\sqrt{2}} + y \cdot \frac{1}{\sqrt{2}} = 3 \]

3. **Multiply through by \( \sqrt{2} \)**:
\[ x + y = 3\sqrt{2} \]

4. **Rearrange to standard form**:
\[ x + y - 3\sqrt{2} = 0 \]

### (ii) For \( p = 5 \) and \( \alpha = 135^\circ \)

1. **Calculate \( \cos 135^\circ \) and \( \sin 135^\circ \)**:
\[ \cos 135^\circ = -\frac{1}{\sqrt{2}}, \quad \sin 135^\circ = \frac{1}{\sqrt{2}} \]

2. **Substitute into the equation**:
\[ x \cdot \left(-\frac{1}{\sqrt{2}}\right) + y \cdot \frac{1}{\sqrt{2}} = 5 \]

3. **Multiply through by \( \sqrt{2} \)**:
\[ -x + y = 5\sqrt{2} \]

4. **Rearrange to standard form**:
\[ -x + y - 5\sqrt{2} = 0 \quad \text{or} \quad x - y + 5\sqrt{2} = 0 \]

### (iii) For \( p = 8 \) and \( \alpha = 150^\circ \)

1. **Calculate \( \cos 150^\circ \) and \( \sin 150^\circ \)**:
\[ \cos 150^\circ = -\frac{\sqrt{3}}{2}, \quad \sin 150^\circ = \frac{1}{2} \]

2. **Substitute into the equation**:
\[ x \cdot \left(-\frac{\sqrt{3}}{2}\right) + y \cdot \frac{1}{2} = 8 \]

3. **Multiply through by 2**:
\[ -\sqrt{3}x + y = 16 \]

4. **Rearrange to standard form**:
\[ -\sqrt{3}x + y - 16 = 0 \]

### (iv) For \( p = 3 \) and \( \alpha = 225^\circ \)

1. **Calculate \( \cos 225^\circ \) and \( \sin 225^\circ \)**:
\[ \cos 225^\circ = -\frac{1}{\sqrt{2}}, \quad \sin 225^\circ = -\frac{1}{\sqrt{2}} \]

2. **Substitute into the equation**:
\[ x \cdot \left(-\frac{1}{\sqrt{2}}\right) + y \cdot \left(-\frac{1}{\sqrt{2}}\right) = 3 \]

3. **Multiply through by \( \sqrt{2} \)**:
\[ -x - y = 3\sqrt{2} \]

4. **Rearrange to standard form**:
\[ x + y + 3\sqrt{2} = 0 \]

### (v) For \( p = 2 \) and \( \alpha = 300^\circ \)

1. **Calculate \( \cos 300^\circ \) and \( \sin 300^\circ \)**:
\[ \cos 300^\circ = \frac{1}{2}, \quad \sin 300^\circ = -\frac{\sqrt{3}}{2} \]

2. **Substitute into the equation**:
\[ x \cdot \frac{1}{2} + y \cdot \left(-\frac{\sqrt{3}}{2}\right) = 2 \]

3. **Multiply through by 2**:
\[ x - \sqrt{3}y = 4 \]

4. **Rearrange to standard form**:
\[ x - \sqrt{3}y - 4 = 0 \]

### (vi) For \( p = 4 \) and \( \alpha = 180^\circ \)

1. **Calculate \( \cos 180^\circ \) and \( \sin 180^\circ \)**:
\[ \cos 180^\circ = -1, \quad \sin 180^\circ = 0 \]

2. **Substitute into the equation**:
\[ x \cdot (-1) + y \cdot 0 = 4 \]

3. **Simplify**:
\[ -x = 4 \]

4. **Rearrange to standard form**:
\[ x + 0y - 4 = 0 \quad \text{or simply} \quad x - 4 = 0 \]

### Summary of Equations
1. \( x + y - 3\sqrt{2} = 0 \)
2. \( x - y + 5\sqrt{2} = 0 \)
3. \( -\sqrt{3}x + y - 16 = 0 \)
4. \( x + y + 3\sqrt{2} = 0 \)
5. \( x - \sqrt{3}y - 4 = 0 \)
6. \( x - 4 = 0 \)