The figure represented on an Argand diagram by this equation is the set of points twice as far from 1 as from i.
Intersection of two cones[]
In x-y-z space, let cone A be given by and cone B be given by The projection of the intersection of these two cones onto the x-y plane is exactly the solution to the student's question. But what about the three-dimensional intersection of two cones? Since its projection onto the x-y plane is a circle, it might be an ellipse. But it can't be an ellipse because it's not a plane figure; it's a kind of saddle shape, conforming to the surface of the larger cone, and wrapped around the body of the smaller cone. More research is needed in this wiki regarding the intersection of two parallel right circular cones.[improvement needed: elaborate]
But what about the three-dimensional intersection of two cones? Since its projection onto the x-y plane is a circle, it might be an ellipse. But it can't be an ellipse because it's not a plane figure; it's a kind of saddle shape, conforming to the surface of the larger cone, and wrapped around the body of the smaller cone.
Solution[]
First, we'll do the specific solution, then a more general solution.
Specific solution[]
Letting z=x+iy, we see that plotting on an argand diagram is the same as
plotting on the x-y plane.
Expanding gives
Multiply through by −3 to give you
Now complete the squares as
So the solution is a circle with center and radius
General solution[]
Now, generalize. Plot Letting and
Rewrite this as
and then expand it, giving
then gather terms on the left, constants on right:
now multiply through by so that the terms resemble