Question

The straight line through P (x₁, y₁) inclined at an angle θ with the x-axis meets the line ax + by + c = 0 in Q. Find the length of PQ.

Answer 1

The equation of the line that passes through \[P \left( x_1 , y_1 \right)\] and makes an angle of \[\theta\] with the x-axis is \[\frac{x - x_1}{cos\theta} = \frac{y - y_1}{sin\theta}\].

Let PQ = r
Then, the coordinates of Q are given by \[x = x_1 + r\text { cos }\theta, y = y_1 + r\text { sin }\theta\]

Thus, the coordinates of Q are \[\left( x_1 + r\text { cos }\theta, y_1 + r\text { sin }\theta \right)\].

Clearly, Q lies on the line ax + by + c = 0.

\[\therefore a\left( x_1 + r\text { cos }\theta \right) + b\left( y_1 + r\text { sin }\theta \right) + c = 0\]

\[ \Rightarrow r = - \frac{a x_1 + b y_1 + c}{a\cos\theta + b\text { sin }\theta}\]

∴ PQ = \[\left| \frac{a x_1 + b y_1 + c}{a\cos\theta + bsin\theta} \right|\]