Find Intersection of Complex Number Loci Given w

In summary, the conversation discusses the problem of finding the intersection of two loci on a complex plane. The first locus is determined by the equation |z-w| = |z-iw|, which represents points equally distant from w and iw. The second locus involves the argument of the complex number z-w and is solved using the multiplication properties of complex numbers. The solution involves finding the perpendicular passing through the midpoint of the two loci and can be approached geometrically or algebraically.
  • #1
Punch
44
0
w is a fixed complex number and \( 0<arg(w)<\frac{\pi}{2} \). Mark A and B, the points representing w and iw, on the Argand dagram. P represents the variable complex number z. Sketch on the same diagram, the locus of P in each of the following cases: (i) \( |z-w|=|z-iw| \) (ii) \(arg(z-w)=arg(iw)\)

Find in terms of w, the complex number representing the intersection of the two loci.

I have drawn the 2 locus already. But I do not know how to find the complex number representing the intersection of the 2 loci.
Do I form the equation of the 2 loci and then find the intersection by substituting one into the other?
 
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  • #2
Use \ ( and \ ) without spaces to make your LaTeX work. As for the problem, remember that when you multiply complex numbers you rotate and expand/contract them, i.e., if \( z_1 = r_1 e^{ix_1} \text{ and } z_2 = r_2e^{i x_2} \text{ then } z_1z_2 = r_1r_2e^{i(x_1+x_2)} \). When you have \( |z-w| \) what you are measuring is the distance between \( z \text{ and } w \). Imposing that \( |z-w| = |z-iw| \) you want the locus of the points that are equally distant from \( w \text{ and } iw \).

Try working the second the same way. Remember the argument is the angle the complex number makes with the real axis.
 
  • #3
Fantini said:
Use \ ( and \ ) without spaces to make your LaTeX work. As for the problem, remember that when you multiply complex numbers you rotate and expand/contract them, i.e., if \( z_1 = r_1 e^{ix_1} \text{ and } z_2 = r_2e^{i x_2} \text{ then } z_1z_2 = r_1r_2e^{i(x_1+x_2)} \). When you have \( |z-w| \) what you are measuring is the distance between \( z \text{ and } w \). Imposing that \( |z-w| = |z-iw| \) you want the locus of the points that are equally distant from \( w \text{ and } iw \).

Try working the second the same way. Remember the argument is the angle the complex number makes with the real axis.

Yup, I think you haven't read the next part I wrote. I completed drawing the locus and am facing difficulties solving the part which asks for a complex number representing the intersection of these 2 loci. "I have drawn the 2 locus already. But I do not know how to find the complex number representing the intersection of the 2 loci.
Do I form the equation of the 2 loci and then find the intersection by substituting one into the other?"
 
  • #4
Geometrically, it will be the perpendicular passing through the midpoint connecting those two. Every point of it is equally distant to both. Algebraically, when you solve \( |z-w| = |z-iw| \) you should get two points, get the line passing through them and that's you answer. Since he asks for a sketch only, the geometric description should be easier to follow.
 
  • #5
Fantini said:
Geometrically, it will be the perpendicular passing through the midpoint connecting those two. Every point of it is equally distant to both. Algebraically, when you solve \( |z-w| = |z-iw| \) you should get two points, get the line passing through them and that's you answer. Since he asks for a sketch only, the geometric description should be easier to follow.
The OP has said a few times now that s/he is NOT having trouble getting each locus, the trouble is getting the intersection of the two loci.

@OP: I have not looked closely, but you might be able to construct the intersection point geometrically in terms of w by using the isosceles triangles and symmetry that is present. Alternatively, an algebraic solution could be hammered out by substituting w = a + ib and z = x + iy into each locus to get the Cartesian equation and then solve using simultaneous equations and then link the answer back to w.
 
  • #6
I'm sorry for not understanding the question properly, when I gave it further analysis I realized I was of no help.
 

Related to Find Intersection of Complex Number Loci Given w

What is the definition of a complex number?

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1. It is typically written as z = a + bi.

What is the locus of a complex number?

The locus of a complex number is the set of points in the complex plane that satisfy a given condition or equation. In other words, it is the collection of all complex numbers that satisfy a specific rule or property.

What is the intersection of complex number loci?

The intersection of complex number loci is the set of points that satisfy all of the given conditions or equations for each locus. In other words, it is the common solution for all of the loci involved.

How do you find the intersection of complex number loci?

To find the intersection of complex number loci, you must first set up and solve a system of equations that represents each locus. The solution to this system will be the coordinates of the intersection points, if any exist.

What are some real-world applications of finding the intersection of complex number loci?

One real-world application of finding the intersection of complex number loci is in engineering, specifically in designing circuits and systems. Another application is in geometry, where it can be used to find the intersection points of geometric figures in the complex plane.

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