A line, m, is parallel to a plane, X, and is 6 inches from X...

A line, $m,$ is parallel to a plane, $X,$ and is 6 inches from $X .$ The set of points that are 6 inches from $m$ and 1 inch from $X$ form
(A) a line parallel to $m$
(B) two lines parallel to $m$
(C) four lines parallel to $m$
(D) one point
(E) the empty set

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Key Concepts

Intersection of Surfaces

The intersection of different geometric surfaces, such as a cylinder and a plane, is a fundamental concept in spatial geometry. When two loci defined by separate distance conditions intersect, the resulting set often consists of curves or lines, and understanding this intersection is key to solving problems involving multiple constraints.

Cylindrical Surfaces

A cylindrical surface is formed by all points in space that are at a fixed distance from a given line (the axis). This concept is applied when describing the set of points that maintain a constant distance from a straight line, leading to a cylindrical shape in three dimensions.

Distance in Three-Dimensional Space

Understanding how distances are measured in three-dimensional geometry is essential, particularly when determining the distance from a point to a line or from a point to a plane. These measurements help establish the relative positions of objects in space and form the basis for defining geometric loci.

Locus of Points

A locus in geometry is the collection of all points that satisfy a particular condition or set of conditions. This concept is crucial when constructing geometric figures from distance constraints, such as all points equidistant from a given line or a plane.

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