MHB Find Intersection of Complex Number Loci Given w

[Punch]

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?

 

[Fantini]

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]

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]

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]

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]

I'm sorry for not understanding the question properly, when I gave it further analysis I realized I was of no help.