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Problem 775

# Describe the locus which is the set of points in a plane lying a distance of 3 units from a line in the plane. What is the locus if we remove the condition that the given points and line lie on the same plane?

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The locus of points in a plane that are 3 units away from a given line consists of two parallel lines that are both 3 units away from the initial line. If we remove the condition that the points must lie on the same plane, the locus becomes two families of parallel planes, each at a distance of 3 units from the initial plane containing the given line, one on each side of the plane.
See the step by step solution

## Step 1: Visualize the situation in the plane

To better understand what the locus looks like, let's draw a line in the plane and some points that are 3 units away from the line. Notice that, at any point along the line, we can form two points that are 3 units away from the line, one on each side of the line. These sets of points will be parallel to the initial line and at a constant distance of 3 units.

## Step 2: Write a description of the locus in the plane

Since the locus consists of all the points that are at a constant distance of 3 units from the given line, the locus is a pair of lines parallel to the initial line and at a distance of 3 units from it.

## Step 3: Visualize the situation without the plane restriction

In order to find the locus when the restriction that the given points must lie in the same plane as the initial line is removed, we have to consider the set of points that are 3 units away from the line in three-dimensional space. The space at 3 units from a line in 3D will form two sets of parallel planes to the initial plane (one on each side of the plane).

## Step 4: Write a description of the locus without the plane restriction

When the condition that the given points and line must lie on the same plane is removed, the locus is the set of points that form two families of parallel planes, one on each side of the initial plane containing the given line. Each plane in the family is parallel to the initial plane, and its distance from the initial plane is equal to 3 units.

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