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Complex Numbers V

Punch

New member
Jan 29, 2012
23
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?
 
Last edited:

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
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.
 

Punch

New member
Jan 29, 2012
23
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?"
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
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.
 

Mr Fantastic

Member
Jan 26, 2012
66
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.
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
I'm sorry for not understanding the question properly, when I gave it further analysis I realized I was of no help.