S.O.S. Mathematics CyberBoard

[zwmurwal]

(1) The point z in the argand diagram moves in such a way that |(2z+1)/(iz +1)|=2. Show that it describes a straight line.
(2) Find all the cube roots of 2i-2

[Soroban]

Hello, zwmurwal!

This is VERY long ... you can understand why I'd rather not post it.

We want the three cube roots of: z = -2 + 2i.

If we plot that complex number on the complex plane,
we have a vector with angle and magnitude

Writing z in "polar" form:


Then

Using Demoivre's formula (or is it Euler's?):


And we've found ONE of the cube roots!
It's the vector to the point (1,1) -- with magnitude sqrt(2).
~~~~~~~~~~~~~~~~~~~~

We know that the three cube roots are spaced equally about the orogin.
So, they are spread radians apart, or

So, we know where the second cube root is --- at the angle

So, we must evaluate: and

We use the Compound angle Formulas:



For example,

. . . or

Then evaluate . . . to get the second cube root.

To get the third cube root, work with the angle

*whew* -- I hope this is clear enough so you can finish the problem.

[Kermie]

[SilverSprite]

[Kermie]

OK, do you understand that |z1 - z2| gives the distance between the points z = z1 and z = z2? If not, sub z1 = x1 + iy1 and z2 = x2 + iy2 into |z1 - z2|. Work through the algebra - you'll end up with the classic expression for the distance between the two points (x1, y1) and (x2, y2). This understanding is the key to the argument.

Right then. Now consider the relation |z - z1| = |z - z2| ....

Geometrically, what it's saying is that the distance of z from z1 is equal to the distance of z from z2.

Now join z1 and z2 by a line segment. Clearly a value of z that will satisfy the given relation is the value of z at the midpoint of this line segment. This is because the distance of the midpoint from z1 is the same as its distance from z2 (by definition of a midpoint!!).

Now what I want you to do is imagine a line passing through the midpoint that is perpendicular to the line segement. Do you see that, by symmetry, every point on this line will be the same distance from z1 and z2? You'll probably need to draw yourself a diagram to see this. So the locus of points defined by |z - z1| = |z - z2| is this straight line.

Now this straight line bisects the line segment AND is perpendicular to the line segment. That is, it is the straight line that is the perpendicular bisector of the line segment joining the points z = z1 and z = z2.

[SilverSprite]

Yeah! I've got it now thanks! OK I understand. Thank you so much kermie.