Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.6 AIME, harder Find the answer

# Task 1.

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 plant producing slabs, located at a distance $d$ from the PL $(d \neq 0)$ . For rhythmic operation, it is required that each point of the road under construction be equally distant from the plant and from the PL. What line on the plane describes the road under construction? Introduce a suitable coordinate system and find the equation of the line describing the road in this coordinate system; determine the type of the line.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

# Solution.

Let's choose a rectangular coordinate system $X O Y$ , placing the concrete slab factory at the origin $O$ of this coordinate system and directing the axis $O X$ along the power line. Let's assume for definiteness that the road passes through the half-plane $\{y \geq 0\}$ . Then the power line will correspond to the line $y=d$ .

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

Equating the squares of these distances:

$x^{2}+y^{2}=|y-d|^{2}$

From this, we obtain the equation of the line of the road being built:

$y=\frac{d}{2}-\frac{1}{2 d} \cdot x^{2}$

This is a parabola.

Answer: The line of the road being built is a parabola. One possible equation is given above.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.