Geometric Arguments for Z1-Z2 in Complex Plane

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In summary, a geometric argument is a way of proving something using visual or physical evidence. In the given problem, the expression |z-4i| + |z+4i|=10 represents an ellipse with foci at (0, 4i) and (0, -4i). This can be shown by considering the locus of points whose distance from these two fixed points sum up to a constant value of 10. Alternatively, the expression can also be rewritten using the definition of distance in the complex plane, |z1-z2|, where z1 and z2 are complex numbers. This leads to the equation (x/a)^2 + (y/b)^2 = 1, which is the standard form of
  • #1
Ed Quanta
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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?
 
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  • #2
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
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
matt grime said:
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.
 
  • #5
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
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.
 

1. What is the concept of geometric arguments for Z1-Z2 in the complex plane?

Geometric arguments for Z1-Z2 in the complex plane refer to using visual representations and properties of geometric shapes to prove the difference between two complex numbers, Z1 and Z2. It is a method of mathematical proof that utilizes the properties and relationships of geometric figures to explain the difference between two complex numbers.

2. How are geometric arguments used to prove the difference between complex numbers in the complex plane?

Geometric arguments use the properties of geometric shapes, such as circles and triangles, to represent and manipulate complex numbers in the complex plane. By utilizing the geometric properties and relationships of these shapes, we can prove the difference between two complex numbers, Z1 and Z2, by showing that the geometric representation of Z1 and Z2 results in a different shape or position in the complex plane.

3. What are some examples of geometric arguments for Z1-Z2 in the complex plane?

Examples of geometric arguments for Z1-Z2 in the complex plane include using the properties of circles to prove the difference between two complex numbers, such as the distance between the centers of two circles representing Z1 and Z2. Another example is using the properties of triangles, such as the Pythagorean theorem, to prove the difference between two complex numbers by showing that the lengths of the sides of the triangle representing Z1 and Z2 are different.

4. What are the advantages of using geometric arguments for Z1-Z2 in the complex plane?

One advantage of using geometric arguments for Z1-Z2 in the complex plane is that it provides a visual representation of complex numbers, making it easier to understand and visualize the concept. Additionally, it allows for a more intuitive and geometric approach to proving the difference between complex numbers, which can be helpful for those who struggle with more abstract mathematical concepts.

5. Are there any limitations to using geometric arguments for Z1-Z2 in the complex plane?

While geometric arguments can be a useful tool for proving the difference between complex numbers in the complex plane, it may not always be applicable or feasible. Some complex numbers may not have a clear geometric representation, making it difficult to use this method. Additionally, geometric arguments may not be the most efficient or accurate method for proving the difference between complex numbers in some cases.

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