# Problem 5.

A road needs to be built, paved with concrete slabs. It will pass through an area where there is a straight section of a power line (PL) and a slab manufacturing plant located at a distance $d$ from the PL $(d \neq 0)$. For rhythmic operation, it is required that each point on the road being built is equally distant from the plant and from the PL.

A) Introduce a coordinate system such that the brick plant has coordinates $(0,0)$, and the PL passes through the point $(0, d)$ parallel to one of the coordinate axes, and find the coordinates of the point on the road that is at a distance of $5 d$ from the plant.

B) For which natural numbers $n$ does such a road have a point that is at a distance of $n d$ from the plant?
# 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 $y \leq d$ ).

A) 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 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 $\sqrt{x^{2}+(-4 d)^{2}}=5 d$.

It has two solutions $\quad x= \pm 3 d$.

B) To find the specified point, we first need to solve the equation

$$
|y-d|=5 d
$$

with the additional condition $y \leq d$. Such a solution always exists and is equal to $-(n-1) d$. Then we get the equation $\sqrt{x^{2}+(n-1)^{2} d^{2}}=n d$, which has the solution $x= \pm \sqrt{2 n-1} d$ for any natural number $n$.

Answer. A) two points $(-3 d,-4 d),(3 d,-4 d)$. B) for any natural numbers $n$.