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 pm beta =(pi)/(2)`
B. `alpha+(pi)/(2)`
C. `|alpha-beta|=(pi)/(2)`
D. `alpha=beta`

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: Identify the equations of the lines
The equations of the lines are given as:
1. \( x \cos \alpha + y \sin \alpha = P_1 \) (Line 1)
2. \( x \cos \beta + y \sin \beta = P_2 \) (Line 2)

### Step 2: Rewrite the equations in slope-intercept form
To find the slopes of the lines, we can rearrange each equation into the form \( y = mx + c \).

For Line 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 Line 1 is:
\[ m_1 = -\frac{\cos \alpha}{\sin \alpha} \]

For Line 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 Line 2 is:
\[ m_2 = -\frac{\cos \beta}{\sin \beta} \]

### Step 3: Use the condition for perpendicular lines
Two lines are perpendicular if the product of their slopes is equal to \(-1\):
\[ m_1 \cdot m_2 = -1 \]
Substituting the values of \( m_1 \) and \( m_2 \):
\[ \left(-\frac{\cos \alpha}{\sin \alpha}\right) \cdot \left(-\frac{\cos \beta}{\sin \beta}\right) = -1 \]
This simplifies to:
\[ \frac{\cos \alpha \cos \beta}{\sin \alpha \sin \beta} = -1 \]

### Step 4: Rearranging the equation
Multiplying both sides by \(\sin \alpha \sin \beta\):
\[ \cos \alpha \cos \beta = -\sin \alpha \sin \beta \]

### Step 5: Use trigonometric identities
Using the trigonometric identity \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \):
\[ \cos \alpha \cos \beta + \sin \alpha \sin \beta = 0 \]
This implies:
\[ \cos(\alpha - \beta) = 0 \]

### Step 6: Solve for the angles
The cosine of an angle is zero when the angle is an odd multiple of \( \frac{\pi}{2} \):
\[ \alpha - \beta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
Taking the absolute value, we can express this as:
\[ |\alpha - \beta| = \frac{\pi}{2} \]

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