
#1
Jun2504, 11:39 AM

P: 297

I am told that z1z2 is the distance between two points z1 and z2 in the complex plane. I have to give a geometric argument that
a) z4i + z+4i=10 represents an ellipse whose foci are (0, and positive or negative 4) b)z1=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? 



#2
Jun2504, 11:54 AM

Emeritus
Sci Advisor
PF Gold
P: 11,154

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. 



#3
Jun2504, 02:30 PM

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P: 9,398

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.




#4
Jun2504, 04:42 PM

Emeritus
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PF Gold
P: 11,154

Geometric arguments 



#5
Jun2504, 05:11 PM

Sci Advisor
HW Helper
P: 9,398

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.




#6
Jun2604, 10:23 AM

Math
Emeritus
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Thanks
PF Gold
P: 38,879

From the way the original question was phrased: "give a geometric argument that
a) z4i + 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: z4i+ 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. 


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