# Complex Numbers V

#### Punch

##### New member
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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#### Fantini

MHB Math Helper
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
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

MHB Math Helper
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
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.