Locus of complek numbers (1 Viewer)

[Aysce]

Q. If z + (1/z) = k, where k is a real number, find the locus of z.

 

[Carrotsticks]

There will be two cases, depending on the value of k:

From the original expression, we acquire the quadratic:



Using the quadratic formula, we have:



Remember that z is a complex number, so it can be real or complex (since the field of is a subfield of )

Hence, we can have 3 cases for the inside of the square root. It can be positive or equal to zero (real) or less than zero (complex):

For the real case:



This implies that:



Hence, the locus if z is real is y=0 or in other words, the x axis.

Now consider the unreal case:





But since the inside of the square root is negative, we will re-write it this way (it's the same thing if you expand the i back inside):



We will now separate them so we can have a very obvious real and unreal component:





Equate the real and unreal components:





We now make k^2 the subject, using the expression we acquired by equating real:



Substitute it back into the expression we had for y (so we are basically eliminating k):



With a bit of manipulation, which will involve squaring both sides, we acquire the expression:



And this is the locus for the unreal case, which is acquired when:



So to finalise our answer, our two possible loci are either the x axis, or the unit circle.

 

[pokka]

 

[Carrotsticks]

^^^ pokka's method is very good. It gives you the locus faster than my method, but I'm not sure how it would determine the range of k such that we would use y=0 or the unit circle.

Also might I add a correction to my solution. Where it says:



It should say:

 

[math man]

In pokka's method a faster way would be from line two and:

 

[largarithmic]

In pokka's method a faster way would be from line two and:
View attachment 24289

Ive got another way which I think is much faster, using real/imaginary parts and modulus/argument and de moivres:




And thus if z+1/z is real, it has imaginary part zero, so either
or .

The first case gives the entire x axis (with the origin removed of course since there 1/z isnt defined), the second gives r = 1 (as r is a positive real) i.e. the unit circle.


Basically Ive done it geometrically. To go from z to 1/z, you reflect it about the x-axis and then change its modulus to the inverse of its modulus. If z+1/z = 0 then, the "height" of z above the x-axis is going to be equal to the "height" of 1/z BELOW the x-axis, and its very easy to see that only happens if the modulus of z and the modulus of 1/z are the same i.e. if z is on the unit circle.

 

[Aysce]

Very nice methods but one question, how do you know which one to take as the locus?

I also have another question if you guys don't mind.

Q. If R is a real number, find the locus of z defined by z = (1+iR)/(1-iR)

Now I multiplied by the conjugate and let z = x+iy where I then equated real and imaginary coefficients. Based on Terry Lee's solution, (Page 52 Q8) it automatically uses x^2 + y^2 when you gain x and y to find the locus. Why do they do this? I don't understand why he did this.