# Points on Cocentric Circles

• Nov 21st 2011, 09:25 PM
tariqchpk
Points on Cocentric Circles
There are two concentric circles with center O

Attachment 22830
Known Points A,B,C,O,M, Radius OC=OA=OB=r1 and OC’=OA’=OB’=r2 are also known
Assumption. M is the mid point of AB and A’B’ , Angle OMB=Angle OMA=90 degree
To be found Points A’, B’, C’ in Cartesian Coordinates
• Nov 22nd 2011, 04:43 AM
Soroban
Re: Points on Cocentric Circles
Hello, tariqchpk!

Quote:

There are two concentric circles with center O.

Attachment 22830

Given: Points $\displaystyle A, B, C, O, M$
. . . . .$\displaystyle OC = OA = OB = r,\;OC’ = OA’ = OB’ = R$
. . . . .$\displaystyle M$ is the midpoint of $\displaystyle AB$ and $\displaystyle A’B’.$
. . . . .$\displaystyle \angle OMB = \angle OMA = 90^o$

Find points $\displaystyle A’, B’, C’$ in cartesian coordinates.

Place center $\displaystyle O$ at the origin.

The larger circle has equation:
. . $\displaystyle x^2 + y^2 \:=\:R^2 \quad\Rightarrow\quad x \:=\:\pm\sqrt{R^2 - y^2}$

Let $\displaystyle m = MO.$
The horizontal line has equation:.$\displaystyle y = m$

The line and the circle intersect at:
. . $\displaystyle A'\left(\sqrt{R^2-m^2},\:m\right)\,\text{ and }\,B'\left(\text{-}\sqrt{R^2-m^2},\:m\right)$

And we have:.$\displaystyle C'(0,\,\text{-}R)$

• Nov 22nd 2011, 06:41 PM
tariqchpk
Re: Points on Cocentric Circles
Dear Soroban

There is a problem in my case i.e. cernter of the circles O is not the origin so what changes be made to accommodate it?
• Nov 23rd 2011, 01:04 AM
mr fantastic
Re: Points on Cocentric Circles
Quote:

Originally Posted by tariqchpk
Dear Soroban

There is a problem in my case i.e. cernter of the circles O is not the origin so what changes be made to accommodate it?

You are expected to make an effort here. If you understand post #2 you should be able to make the necessary changes yourself.