Question

The straight line through $(P{x}_1,y_1)$ inclined at an angle $\theta$ with the x-axis meets the line ax + by + c = 0 in Q. Find the length of PQ.

Answer 1

$\text{The equation of the line that passes through}P(x_1,y_1)\text{and makes an angle of}$

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

$\text{Let PQ = r}$

$\text{Thus, the coordinates of Q are}(x_1+rcos\theta ,y_1+rsin\theta )$

$x=x_1+rcos\theta ,y=y_1+rsin\theta$

$\text{Clearly, Q lies on the lines}ax+by+c=0$

$\therefore a(x_1+rcos\theta )+b(y_1+rsin\theta )+c=0$

$\Rightarrow r=-\frac{a{x}_1+b{y}_1+c}{acos\theta +bsin\theta }$

$\therefore PQ=\left|\frac{a{x}_1+b{y}_1+c}{acos\theta +bsin\theta }\right|$