MHB Can Three Circles Intersect at a Common Point?

AI Thread Summary
The discussion focuses on solving the problem of finding the intersection point of three circles defined by their equations. The circles' equations are provided, and a method involving subtraction of these equations is used to eliminate squares and simplify the problem. Through this process, the intersection point (3, 1) is determined, along with an additional intersection point (11/5, 13/5). The conversation highlights the mathematical techniques used, including matrix manipulation and factoring quadratic equations. Overall, the thread demonstrates a systematic approach to solving the intersection of multiple circles.
karush
Gold Member
MHB
Messages
3,240
Reaction score
5
studying with a friend there was the intersection of 3 circles problem which is in common usage
here is my overleaf output
View attachment 9075

I was wondering if this could be solved with a matrix in that it has squares in it

or is there a standard equation for finding the intersection of 3 circles given the centers and radius'
and an assumed intersection
 
Mathematics news on Phys.org
Yes, the three circles have equations
(x- 7)^2+ (y- 4)^2= 25
(x+ 9)^2+ (y+ 4)^2= 169 and
(x+ 3)^2+ (y- 9)^2= 100

Multiplying those squares gives
x^2- 14x+ 49+ y^2- 8y+ 16= 25
x^2+ 18x+ 81+ y^2+ 8y+ 16= 169
x^2+ 6x+ 9+ y^2- 18y+ 81= 100

And subtracting will get rid of the squares!

Subtracting the first equation from the second gives
32x+ 32+ 16y= 144
Subtracting the first equation from the third gives
20x- 40- 10y+ 65= 75.<br /> <br /> 32x+ 16y= 112 so 2x+ y= 7<br /> 20x- 10y= 50 so 2x- y= 5.<br /> Adding those 4x= 12 so x= 3 and then y= 1. <br /> <br /> That, (3, 1), is the point where all three circles intersect.<br /> <br /> We also can look at 2x+ y= 7, so y= 7- 2x and (x- 7)^2+ (y- 4)^2= (x- 7)^2+ (3- 2x)^2= x^2- 14x+ 49+ 9- 12x+ 4x^2= 5x^2- 26x+ 58= 25. 5x^2- 26x+ 33= 0. That can be factored as (5x- 11)(x- 3)= 0so x= 3 or x= 11/5. If x= 3 y= 7- 6= 1 and if x= 11/5, y= 7- 22/5= (35- 22)/5= 13/5. (3, 1) and (11/5, 13/5) is another intersection.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top