Complex variables : Triangle inequality

In summary, The equation |a-4*i| + |z+4*i| =10 represents an ellipse with foci at (0,4) and (0,-4), using the triangle inequality and distance formula to calculate the distance between the two points and the foci. The variable 10 in the equation represents the distance between the points on the ellipse's contour, not the distance between the foci.
  • #1
Benzoate
422
0

Homework Statement



Using the fact that |z(1)-z(2)| is the distance between two points z(1) and z(2) , give a geometric argument that

a)|a-4*i| + |z+4*i| =10 represents an ellipse whose foci are (0,4) and(0,-4).

Homework Equations



Triangle inequality equation; distance formula


The Attempt at a Solution



|z(1)-z(2)|=sqrt((0-0)^2 + (4-(-4))^2)= 8 . How is the Radius =10 of the equation related to the distance between z(1) and z(2) which I calculated to be 8. If there is a relationship between the distance formula and the radius, how will the relationship between the radius and the distance between the two points help me determined if|z-4i| + |z+4i|=10 represents an ellipse with foci (0,4) and (0,-4).
 
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  • #2
You calculated the distance between the foci didn't you? The foci don't lie on the contour of the ellipse. Besides how are you defining radius for an ellipse?
 
  • #3
NoMoreExams said:
You calculated the distance between the foci didn't you? The foci don't lie on the contour of the ellipse. Besides how are you defining radius for an ellipse?

I don't think there is a radius for an ellispse. But what variable of the ellipse is 10 supposed to represent? distance between the foci is 8.
 

FAQ: Complex variables : Triangle inequality

1. What is the Triangle Inequality Theorem in complex variables?

The Triangle Inequality Theorem in complex variables states that for any two complex numbers, the absolute value of the sum of the two numbers is less than or equal to the sum of the absolute values of each individual number. In other words, the length of the hypotenuse of a triangle cannot be greater than the sum of the lengths of the other two sides.

2. How is the Triangle Inequality Theorem used in complex analysis?

The Triangle Inequality Theorem is used in complex analysis to prove the convergence and continuity of functions, as well as to establish bounds for complex numbers. It is also used in the proof of the Cauchy-Schwarz inequality and the Schwarz lemma.

3. What is the significance of the Triangle Inequality Theorem in mathematics?

The Triangle Inequality Theorem is significant in mathematics because it is a fundamental property of complex numbers and is used extensively in various fields such as complex analysis, geometry, and functional analysis. It also has important applications in physics and engineering.

4. Can the Triangle Inequality Theorem be extended to more than two complex numbers?

Yes, the Triangle Inequality Theorem can be extended to any finite number of complex numbers. This is known as the Generalized Triangle Inequality. It states that for any complex numbers z1, z2, ..., zn, the absolute value of the sum of these numbers is less than or equal to the sum of the absolute values of each individual number.

5. How is the Triangle Inequality Theorem related to the concept of a metric space?

The Triangle Inequality Theorem is closely related to the concept of a metric space. In a metric space, a distance function is defined between any two points, and the Triangle Inequality Theorem is a fundamental property of this distance function. It states that the distance between any two points in a metric space is always less than or equal to the sum of the distances between those points and a third point. This is analogous to the Triangle Inequality Theorem in complex variables, where the distance between two complex numbers is less than or equal to the sum of the distances between the individual numbers and the origin.

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