Distances in analytic geometry

The case of distances deserves a note of its own, because it recurs continually in analytic geometry problems.

Remark — The case of distances in analytic geometry

When a distance appears in a problem, such as $d(P,Q)=\sqrt{(x_Q-x_P)^2+(y_Q-y_P)^2}$, the quantity under the root is a sum of squares, hence it is always $\ge 0$ — even when the coordinates of the points depend on a parameter. No existence conditions are needed on the root of the distance: one squares directly without ever worrying about it.

This small observation saves a lot of time in analytic geometry exercises: every time a distance is compared with a value, one is automatically in a “shortcut” case.

Links

Topics: Irrational inequalities
Concepts: Existence conditions · Irrational inequality · Point–line distance · Line
Methods: Non-negative sides shortcut
Skills: Analytic geometry