Can someone help with Argand plane graphing and finding the radius?

[AlchemistK]

I'm having trouble with the various geometrical representation of complex numbers.
Can someone provide me link where this is discussed, or maybe an argand plane graphing calculator online?

 

[Mentallic]

Is there anything in particular that's troubling you or is it a general problem you're having? Are you studying from a textbook or in school?

 

[AlchemistK]

I'm having trouble with visualizing " |z-z1| = k |z-z2|" where k is an real number. I know that it gives a circle, but where are the complex numbers z1 , z2 located on this circle? What deterrence do the different values of K make? (less than 0, more than 0, less than 1,etc.)

That is just one problem I came across in my book, but I'd love to see all the other different geometrical shapes and representations in the argand plane.

 

[AlchemistK]

Where do z1 and z2 lie with respect to the circle that is formed? What complex number is the center?

 

[chiro]

Where do z1 and z2 lie with respect to the circle that is formed? What complex number is the center?

Hey AlchemistK.

Try expanding the equation until you get the equation for a circle.

Consider that z = a + ib. What you want to do is get an equation in terms of the x and y coordinates, but in terms of an ellipse or circle: a circle has the equation (x-a)^2 + (y-b)^2 = r^2 for a circle centred at (a,b) with a radius r.

So arrange your equation in terms of your z1 and z2 by making them constant (z1 = c + di, z2 = e + fi) and solve in terms of your z (make z = a + bi). You're a and b terms will be variable (like say x and y in a normal cartesian function) and the c,d,e,f terms are just constants.

Rearrange them so that you an equation like (t - a)^2 + (u - b)^2 = r^2 for some constants t,u, and r where a and b correspond to the z = a + bi representation of z.

If you do this you will understand all the concepts and it will help you with later mathematics.

 

[AlchemistK]

OK on taking z=x+iy, z1 = a+ib and z2= c+id and solving I get :

x^2 + y^2 + 2x[(a - c*k^2)/(k^2 -1)] + 2y[(b - d*k^2 )/(k^2 -1)] - R

Where R is some constant. So the center comes out to be :

[(c*k^2 -a)/(k^2 -1)] , [(d*k^2 -b)/(k^2 -1)]

 

[AlchemistK]

For K greater than 1, the center lies closer to z2 and the other way around for z1.

So I can compare which one lies closer and hence should be in the interior, but I can't calculate the radius (very complex to solve) so I can't tell if there will be cases when both z1 and z2 are inside/outside/on the circle.