# Is this property of parabola true?

Attached text (taken from "Pre Calculus Mathematics in a Nutshell bu George F Simmons) defines a parabola as containing a "property that each of its points is equidistant from a given fixed point and a given fixed line" : But it appears that the fixed point(F) and directrix distance varies at each point. As each point on the parabola is at a different distance in relation to the fixed point ? Or am I not interpreting the statement correctly ?

• You are right the equidistance is the property but the distance varies from point to point and is minimum at the vertex of the parabola – marwalix Feb 14 '15 at 8:31
• A parabola is a locus of points whose perpendicular distance from a fixed line (directrix) is equal to the diatance from some fixed point (focus). – AvZ Feb 14 '15 at 8:32
• @marwalix so text in book is false ? – blue-sky Feb 14 '15 at 8:44
• No, you are confusing equidistant and fixed distance. Your textbook never mentioned the latter ... So reading it carefully, your textbook is correct and your interpretation that the textbook should state that the distance is fixed is wrong! – String Feb 14 '15 at 8:51
• Given the distance from the point, the distance from the line must be the same. It certainly varies as the point moves along the parabola. – MPW Feb 14 '15 at 8:54

## 2 Answers

The phrase "each of its points" is singular, referring to an individual point. Each point must, all by itself, be equidistant from the focus and the directrix. The figure depicts one such point.

There's no requirement in this definition that compares the distances to different points on the parabola. You're broadly right that "each point on the parabola is at a different distance"; that doesn't conflict with the definition.

Let $P_1,P_2,P_3,\ldots,P_n$ be points along the parabola. Let $F_1,F_2,F_3,\ldots,F_n$ be the distances from each such point to the focus F, and $D_1,D_2,D_3,\ldots,D_n$ be the distances from each such point to the directrix D. What the book is saying is that $F_k=D_k$ for each $1\le k\le n$, not that $F_k=F_j$ and $D_k=D_j$ for all j and k between $1$ and n.