AB is any diameter of the circle $x^2 + y^2 + 2x + 2y = 0 $ and CD is any diameter of circle $x^2 + y^2 + x + y - 4 = 0 $ such that A, B , C and D are con cyclic. The locus of center of the circle passing through A,B,C,D is ...

Question

mathslover · Answer

@ganeshie8

mathslover · Answer

It's not a big deal. Here is the solution, but, I want someone to explain me that.

mathslover · Answer

I'm not able to understand that from where did the radical axis come? I know that they have used that as a trick, but if there is a possible proof, then I would love to see it.

ganeshie8 · Answer

when circles are tangent to each other, radical axis is the common tangent
i still don't see how this will be the center of circle passing through A,B,C,D

mathslover · Answer

Exactly! :'(

mathslover · Answer

Wait, lemme see what the graph says http://www.wolframalpha.com/input/?i=x%5E2+%2B+y%5E2+%2B+2x+%2B+2y+%3D+0+and+x%5E2+%2B+y%5E2+%2B+x+%2B+y+-+4+%3D+0

mathslover · Answer

Now, let us mark the diameters and all using geogebra. May be that will help us a bit.

mathslover · Answer

$x^2 + y^2 + 2x + 2y = 0 $

Center = (-1,-1) 
Radius = $\sqrt{2}$ 

x^2 + y^2 + x + y - 4 =0 

Center = $(-1/2 , -1/2 )$ 
Radius = $3/\sqrt{2}$

mathslover · Answer

Looks like this.

mathslover · Answer

Well, I will see what I can do with diameters now.

myininaya · Answer

what does concylic mean?

myininaya · Answer

they lie on a common circle ?

mathslover · Answer

Yeah!

ganeshie8 · Answer

I am not able to visualize how that straight line gives the locus of center of circles passing through points A,B,C,D

ganeshie8 · Answer

*radical axis

ganeshie8 · Answer

can you draw atleast one circle that contains the diameter endpoints A,B,C,D ?

mathslover · Answer

Yeah, one minute. I will try.

mathslover · Answer

Interesting.

ganeshie8 · Answer

also draw the locus line from solution

ganeshie8 · Answer

does the center of that circle fall on that locus line ?

mathslover · Answer

I wonder how to draw the center of that circle.

mathslover · Answer

I will try, one sec.

mathslover · Answer

http://web.geogebra.org/app/?id=qEuAL9Nx Well, can you do that on geogebra?

mathslover · Answer

I've shared the link here. I am not sure how will we be able to mark the center of that circle.

ganeshie8 · Answer

it is easy to eyeball actually
it is 1 km away from the locus line in solution

mathslover · Answer

Aw... sorry, I've never used geogebra that much. No idea what is eyeball :/

ganeshie8 · Answer

|dw:1421163193941:dw|

ganeshie8 · Answer

the centers of the circles that pass through points A,B,C,D must lie on that radical axis, yes ?

ganeshie8 · Answer

as per your solution ^

mathslover · Answer

Yes!

mathslover · Answer

Oops... light cut-off! Its raining here... :'( 

I need to sign out now bhaiya! Sorry.

Can we discuss 'bout this tomorrow?

mathslover · Answer

(if you don't mind).

ganeshie8 · Answer

yea sure :)

anonymous · Answer

lala