The lines `x cos alpha + y sin alpha = P_1 and x cos beta + y sin beta = P_2` will be perpendicular, if :

Options

A. `alpha= beta`
B. `|alpha - beta | = pi//2`
C. `alpha = pi//2`
D. `alpha pm beta = pi//2`

Answer

To determine the condition under which the lines \( x \cos \alpha + y \sin \alpha = P_1 \) and \( x \cos \beta + y \sin \beta = P_2 \) are perpendicular, we can follow these steps:

### Step 1: Find the slopes of the lines
The given lines can be rewritten in the slope-intercept form \( y = mx + c \) to find their slopes.

1. For the first line \( x \cos \alpha + y \sin \alpha = P_1 \):
\[ y \sin \alpha = P_1 - x \cos \alpha \]
\[ y = -\frac{\cos \alpha}{\sin \alpha} x + \frac{P_1}{\sin \alpha} \]
Thus, the slope \( m_1 \) of the first line is:
\[ m_1 = -\frac{\cos \alpha}{\sin \alpha} \]

2. For the second line \( x \cos \beta + y \sin \beta = P_2 \):
\[ y \sin \beta = P_2 - x \cos \beta \]
\[ y = -\frac{\cos \beta}{\sin \beta} x + \frac{P_2}{\sin \beta} \]
Thus, the slope \( m_2 \) of the second line is:
\[ m_2 = -\frac{\cos \beta}{\sin \beta} \]

### Step 2: Apply the condition for perpendicular lines
For two lines to be perpendicular, the product of their slopes must equal \(-1\):
\[ m_1 \cdot m_2 = -1 \]

Substituting the slopes we found:
\[ \left(-\frac{\cos \alpha}{\sin \alpha}\right) \cdot \left(-\frac{\cos \beta}{\sin \beta}\right) = -1 \]
This simplifies to:
\[ \frac{\cos \alpha \cos \beta}{\sin \alpha \sin \beta} = -1 \]

### Step 3: Rearranging the equation
Multiplying both sides by \(\sin \alpha \sin \beta\):
\[ \cos \alpha \cos \beta = -\sin \alpha \sin \beta \]

### Step 4: Use the cosine of angle difference identity
Using the identity \( \cos A \cos B + \sin A \sin B = \cos(A - B) \):
\[ \cos \alpha \cos \beta + \sin \alpha \sin \beta = 0 \]
This can be rewritten as:
\[ \cos(\alpha - \beta) = 0 \]

### Step 5: Solve for the angles
The cosine of an angle is zero at odd multiples of \(\frac{\pi}{2}\):
\[ \alpha - \beta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
For the simplest case, we can take:
\[ \alpha - \beta = \frac{\pi}{2} \]

### Conclusion
Thus, the lines will be perpendicular if:
\[ \alpha - \beta = \frac{\pi}{2} \]