# Task 5.

A road paved with yellow bricks needs to be constructed. It will pass through an area where there is a straight section of a power line (PL) and a brick factory located at a distance $d$ from the PL $(d \neq 0)$. For rhythmic work, it is required that each point on the road being built is equally distant from the factory and from the PL. Introduce a coordinate system such that the brick factory has coordinates $(0,0)$, and the PL passes through the point $(0, d)$ parallel to one of the coordinate axes. Find the coordinates of the point on the road that is at a distance of $5 d$ from the factory.
# Solution.

Let the power line run parallel to the coordinate axis $O X$. Then it will correspond to the line $y=d$. Note that the road being built will be located below this line, (in the half-plane $y \leq d$ ).

If $M(x, y)$ is the point on the road being built, then the distance to the factory $M O$ is $\sqrt{x^{2}+y^{2}}$. This distance is equal to the distance between the point $M$ and the power line, which is $|y-d|$.

According to the condition, $|y-d|=5 d$. From the two solutions of the obtained equation, we choose the one that satisfies the condition $y \leq d$. Thus, $y=-4 d$.

It remains to solve the equation $\sqrt{x^{2}+(-4 d)^{2}}=5 d$.

This equation has two solutions $x= \pm 3 d$.

Answer. Two points $(-3 d,-4 d),(3 d,-4 d)$.