The straight line passing through `P(x_1,y_1)` and making an angle `alpha` with x-axis intersects `Ax + By + C=0` in Q then `PQ` =________________

Explanation

To solve the problem of finding the distance \( PQ \) where \( P(x_1, y_1) \) is a point through which a line makes an angle \( \alpha \) with the x-axis and intersects the line \( Ax + By + C = 0 \) at point \( Q \), we can follow these steps:

### Step-by-Step Solution:

1. **Equation of the Line through Point P**:
The line passing through point \( P(x_1, y_1) \) making an angle \( \alpha \) with the x-axis can be represented in parametric form as:
\[ \frac{x - x_1}{\cos \alpha} = \frac{y - y_1}{\sin \alpha} = r \]
Here, \( r \) is the parameter representing the distance from point \( P \) to point \( Q \).

2. **Expressing x and y in terms of r**:
From the parametric equations, we can express \( x \) and \( y \) as:
\[ x = r \cos \alpha + x_1 \quad \text{(1)} \]
\[ y = r \sin \alpha + y_1 \quad \text{(2)} \]

3. **Substituting x and y into the Line Equation**:
We substitute equations (1) and (2) into the line equation \( Ax + By + C = 0 \):
\[ A(r \cos \alpha + x_1) + B(r \sin \alpha + y_1) + C = 0 \]
Expanding this gives:
\[ Ar \cos \alpha + Ax_1 + Br \sin \alpha + By_1 + C = 0 \]
Rearranging terms, we have:
\[ r(A \cos \alpha + B \sin \alpha) + (Ax_1 + By_1 + C) = 0 \]

4. **Solving for r**:
From the above equation, we can isolate \( r \):
\[ r(A \cos \alpha + B \sin \alpha) = - (Ax_1 + By_1 + C) \]
Thus,
\[ r = \frac{-(Ax_1 + By_1 + C)}{A \cos \alpha + B \sin \alpha} \]

5. **Finding the Distance PQ**:
The distance \( PQ \) is equal to \( r \), so we have:
\[ PQ = \left| \frac{-(Ax_1 + By_1 + C)}{A \cos \alpha + B \sin \alpha} \right| \]
This can be simplified to:
\[ PQ = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \cdot \frac{1}{\sqrt{\cos^2 \alpha + \sin^2 \alpha}} = \frac{|Ax_1 + By_1 + C|}{A \cos \alpha + B \sin \alpha} \]

### Final Result:
Thus, the distance \( PQ \) is given by:
\[ PQ = \frac{|-(Ax_1 + By_1 + C)|}{A \cos \alpha + B \sin \alpha} \]