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 $\mathrm{\theta }$ with the x-axis is $\frac{x-x_1}{\mathrm{cos\theta }}=\frac{y-y_1}{\mathrm{sin\theta }}$.

Let PQ = r
Then, the coordinates of Q are given by

$x=x_1+r\mathrm{cos\theta },y=y_1+r\mathrm{sin\theta }$

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

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

$$\begin{gathered} \therefore a\left(x_1+r\mathrm{cos\theta }\right)+b\left(y_1+r\mathrm{sin\theta }\right)+c=0 \\ \Rightarrow r=-\frac{a{x}_1+b{y}_1+c}{a\cos \mathrm{\theta }+b\mathrm{sin\theta }} \end{gathered}$$

∴ PQ = $\left|\frac{a{x}_1+b{y}_1+c}{a\cos \mathrm{\theta }+b\mathrm{sin\theta }}\right|$