Question

Find the set of points the difference of whose distances from two intersecting straight lines is equal to a constant quantity.

Answer 1

Let the two lines be $L_1:ax+by+c=0$ and $L_2:px+qy+r=0$.

Let $P:(x_1,y_1)$ be a point in the locus.

distance of $P$ from $L_1$ is $\left|\frac{a{x}_1+b{y}_1+c}{\sqrt{a^2+b^2}}\right|$

distance of $P$ from $L_2$ is $\left|\frac{p{x}_1+q{y}_1+r}{\sqrt{p^2+q^2}}\right|$

Let the constant difference between these distances be $K$

The locus of the points is given by $\left|\frac{ax+by+c}{\sqrt{a^2+b^2}}\right|-\left|\frac{px+qy+r}{\sqrt{p^2+q^2}}\right|=K$