I am told that |z1-z2| is the distance between two points z1 and z2 in the complex plane. I have to give a geometric argument that a) |z-4i| + |z+4i|=10 represents an ellipse whose foci are (0, and positive or negative 4) b)|z-1|=|z+i| represents the line through the origin whose slope is -1 Now my question is what exactly is a geometric argument, and what is sufficient in showing what I am told to show?
Rewrite the given expression using the language where |z1 - z2| is replaced by the words, "the distance between z1 and z2." a) Compare the statement this generates with the geometrical definition of an ellipse. b) Recall the locus that is found to be a perpendicular bisector.
alternatively use the fact that (rather than hand waving arguments) |z|= sqrt(x**2+y**2) where z=x+iy is a complex number, x,y real.
|z+4i| + |z-4i| = 10 means that the locus of z is the set of points each of whose sum of distances from two fixed points (4i, -4i) is a constant (=10). Is this not just the same as showing that (x,y) satisfy (x/a)^2 + (y/b)^2 = 1. I don't see how it is any less rigorous, and definitely disagree with your description of it as hand waving. tell me where I'm wrong.
when you get round to demonstrating that circles and straight lines are sent to circles and straight lines under mobius transformations you'll appreciate the necessity of the algebraic arguments, though i will agree hand waving is too dismissive.
From the way the original question was phrased: "give a geometric argument that a) |z-4i| + |z+4i|=10 represents an ellipse whose foci are (0, and positive or negative 4)" it's clear (to me, anyway!) that Gokul43201's idea: |z-4i|+ |z+4i|= 10 means that the total distance from z to 4i and -4i is 10: precisely the definition of ellipse, is the intended solution.