Hi
Show that the locus of a point, which moves so as always to be three times further from one fixed point than from another fixed point, is a circle.
How exactly do I do this?
Hi
First it is essential to draw the situation in order to get an idea of the way to proceed.
Let A and B be the two fixed point.
Let M be a point such as AM = 3 BM.
The aim of the exercise is to find the locus of M.
is equivalent to or
Let G be the point defined by then
Develop and simplify to get the answer ...
What you have here is the locus definition of a circle known as a Circle of Apollonius (also known as an Apollonian Circle). Here are some hyperlinks
Circles of Apollonius
Apollonius' Lonely Circle
Locus of Points in a Given Ratio to Two Points from Interactive Mathematics Miscellany and Puzzles
You will find a proof in the Appendix of this paper: Circle of Apallonius
Hello, xwrathbringerx!
Looks like you need a walk-through . . . hints don't seem to work.
Show that the locus of a point, which moves so as always to be
three times further from one fixed point than from another fixed point, is a circle.
Since this claim is made for any two points,
. . place the points on the x-axis, symmetric to the origin.Let be any point such that: .Code:| P | o (x,y) | * * * | * * | * - - o - - - - + - - - - o - - (-a,0) | (a,0) A | B |
Using the Distance Formula: .
Then: .
Square both sides: .
Expand: .
Simplify: .
Divide by 8: .
Complete the square: .
. . And we have: .
This is a circle with center and radius