Expert: Vishwas K Hajirnis - 10/2/2006

Question
I don't understand how I should solve this math problem, and how to
state the conclusion.

What is the locus in space of points that are 5 units from a given
point? 5 units from a given line?

I also have to draw a diagram for this problem, so how should I
determine what is '5 units'?  

Answer
(First the units part of the question.)
Some of the commonly used units of distance are centimeter, inch etc. Now since none of these units have been specifically mentioned in the question, it means you have the freedom to choose any one that is convenient to you. This is because the answers are not going to change if you replace '5 units' with '5 centimeters' or '5 inches' ( or '5 miles' / '5 light years') etc. But since you have been asked to draw diagrams as part of the answer, you should choose a unit of distance and specify it in the answer. (This choice should be easy depending on the country you live. )  

The locus in space of points that are 5 units from a given point?

Surface of a sphere.

( All the points on the surface of a sphere are at a fixed distance from the center. )

The locus in space of points that are 5 units from a given line ?

A surface of a cylinder.
( The given line will be axis of the cylinder. But since is infinitely long, so will be the cylinder. )

Since both the answers are 3 dimensional objects, I think you should not worry about the figures part of the answer. as in any case we can only draw projections of them on the paper. e.g. For a sphere, just draw a circle, do some shading if you are good at drawing, mark the center. etc.