Points that are symmetric with respect to a circle C

[brcole]

Homework Statement

Lemma 1: Fix the circle C with center (x nought, y nought); y nought is greater than 0 and
radius R is less than y nought. Consider two points P (x nought, y noight tilde) and P prime (x nought, -y nought tilde) which are symmetric with respect to x-axis by construcion.

Prove that P and P prime are also symmetric with respect to Circle C if and only if y nought tilde is equal to the sqrt (y nought squared - radius squared).

Homework Equations

on the attachment

The Attempt at a Solution

I believe that these two points are orthogonal with respect to the x-axis. and they are symmetric because they fall on the same ray.

 

[tiny-tim]

Welcome to PF!

Homework Statement

Lemma 1: Fix the circle C with center (x nought, y nought); y nought is greater than 0 and
radius R is less than y nought. Consider two points P (x nought, y noight tilde) and P prime (x nought, -y nought tilde) which are symmetric with respect to x-axis by construcion.

Prove that P and P prime are also symmetric with respect to Circle C if and only if y nought tilde is equal to the sqrt (y nought squared - radius squared).

Homework Equations

on the attachment

Hi brcole! Welcome to PF!

You're making this too complicated …

all the points are on the same "vertical" line …

so just give every point a name, don't use || of vectors, just talk about the length of lines …

Call (x₀,0) Q.

Then you have to prove that CP.CP´ = r²,

and you know what CQ and QP and QP´are …