Spec '10 - Help forum

[Chavi]

Could I please get some help with a few questions in regards to complex numbers/graph sketching?

1. Find the solutions to the equation in polar form. (can this be done *without* solving algebraically first?)

2. Find the square roots of by cartesian methods and hence find exact values for and

3. If, with an Argand diagram with origin O, the point P represent and Q represents , prove that O,P and Q are collinear and find the ratio of OP : OQ in terms of .

4. Find the coordinates of the point of intersection of the following loci: and

Thanks in advance.

[TrueTears]

1. Let u =z^2

2. z^2 = 1+i, solve using de moivers theorem and then change into cartesian form, rest is trivial.

3. let z = x+yi and z conjugate = x-yi, sub it in, simplify and u get an obvious result.

[Chavi]

1. is there a way to do this directly into polar form, without solving for z, and then converting each answer?

2.  the question asked to solve without using de moivers theorem - how would this be done algebraically?

3. I tried that - z = x + iy, and 1/z conju = (x + iy)/(x^2 + y^2)  - but how do i prove they're collinear, and get the ratio?

[TrueTears]

1. yes sub in u = z^2, factorise then use de moivres.

2. z^2-1-i = 0

quadratic formula?

3. z = x+yi 1/z = z/(x^2+y^2)

so the ratio is 1/(x^2+y^2)

[Mao]

1. is there a way to do this directly into polar form, without solving for z, and then converting each answer?

2.  the question asked to solve without using de moivers theorem - how would this be done algebraically?

3. I tried that - z = x + iy, and 1/z conju = (x + iy)/(x^2 + y^2)  - but how do i prove they're collinear, and get the ratio?

1. no, that would get extremely over-complicated. You will need to perform the substitution

2. let , now you have a set of simultaneous equations and

3. colinear: , where k is some constant. In this case, , which is a constant. The ratio OP:OQ is

@TT: using the quadratic formula would land him back in pretty much the same spot:

[TrueTears]

expand

[Mao]

expand

I doubt many specialist students know how to do that, considering it's not really covered in the course.

[Assuming you are referring to ]

[TrueTears]

tis covered in methods, binomial theorem

[kamil9876]

That binomial theorem is A LOT different.

[TrueTears]

same principle, just extend it.

[kamil9876]

oh right, so you mean that you have a proof of the generalized binomial using only the "same principle".

[TrueTears]

it is just extended to C

maybe u can enlighten me with the proof, but i am happy to just apply it for now, will look at the proof once i know how to utilise it.

[kamil9876]

the word "just" doesn't do the difficulty of that any justice.

edit: how about Fermat's last theorem, can I "just" extend that so that x,y,z are any real numbers.

[TrueTears]

sometimes it is good to prove after you apply. after u get the confidence of seeing how it works in action
rmb?

edit: i am talking about applying it, u said urself u didnt even know why it applies over C.

[Mao]

it is just extended to C

maybe u can enlighten me with the proof, but i am happy to just apply it for now, will look at the proof once i know how to utilise it.

Care to show us how it is applied? The only binomial theorem generalised over C that I know of involves an infinite series, which will give the right answer in a very round-about way.