Login | Register
Enrol now for our new online tutoring program. Learn from the best tutors. Get amazing results. Learn more.

Welcome, Guest. Please login or register.

April 15, 2021, 05:38:16 pm

Author Topic: Help with Complex number Loci questions  (Read 512 times)  Share 

0 Members and 1 Guest are viewing this topic.

nat_21

  • Fresh Poster
  • *
  • Posts: 3
  • Respect: 0
Help with Complex number Loci questions
« on: March 05, 2021, 10:24:31 pm »
0
Hi all,
I tried to expand the algebra but it does not work, maybe I went wrong somewhere. Could anyone show me how to solve this question? (IMG_3199)
For example 3.13, how do you find the radius? (IMG_3200)
Thank you very much.

fun_jirachi

  • MOTM: AUG 18
  • Moderator
  • Forum Leader
  • *****
  • Posts: 879
  • All doom and Gloom.
  • Respect: +557
Re: Help with Complex number Loci questions
« Reply #1 on: March 05, 2021, 11:55:31 pm »
+1
Hey :D

For the first question, if you're sure you went wrong with your algebra but that your method is correct, it might be easier for you to post your working and for us to have a look.

For the second question, the following two circle geometry theorems might be helpful to consider:
1. The angle subtended by two points on the circumference to another point on the circumference is half that of the same two points at the centre of the circle.
2. A line from the centre of the circle will bisect any chord at a right angle.

Using these with a little algebra should allow you to find the centre reasonably easily.

Hope this helps :)
Spoiler
HSC 2018: Mod Hist [88] | 2U Maths [98]
HSC 2019: Physics [92] | Chemistry [93] | English Adv [87] | 3U Maths [98] | 4U Maths [97]
ATAR: 99.05

UCAT: 3310 - VR [740] | DM [890] | QR [880] | AR [800]
Guide Links:
Subject Acceleration (2018)
UCAT Question Compilation/FAQ (2020)
Asking good questions

nat_21

  • Fresh Poster
  • *
  • Posts: 3
  • Respect: 0
Re: Help with Complex number Loci questions
« Reply #2 on: March 06, 2021, 03:17:49 pm »
0
Hey :D

For the first question, if you're sure you went wrong with your algebra but that your method is correct, it might be easier for you to post your working and for us to have a look.

For the second question, the following two circle geometry theorems might be helpful to consider:
1. The angle subtended by two points on the circumference to another point on the circumference is half that of the same two points at the centre of the circle.
2. A line from the centre of the circle will bisect any chord at a right angle.

Using these with a little algebra should allow you to find the centre reasonably easily.

Hope this helps :)
Hi, thank you very much, I understand the second question now. For the first question, I am not too sure did I approach it in the right way.  But here is my working, I didn't finish it because I don't know what to do with it. I really appreciate your help.

fun_jirachi

  • MOTM: AUG 18
  • Moderator
  • Forum Leader
  • *****
  • Posts: 879
  • All doom and Gloom.
  • Respect: +557
Re: Help with Complex number Loci questions
« Reply #3 on: March 06, 2021, 04:16:26 pm »
+1
I think from the outset your method is wrong - you're assuming that \(w\) is purely imaginary then finding the locus some \(z\) such that \(z=x+iy\). It should be done in the opposite way ie. you choose some \(z\) such that \(z=x+iy\) and hence determine the conditions on \(x, y\) (ie. the locus!) such that \(w\) is imaginary.

Realising the denominator immediately doesn't help too much - we ideally want to express w in terms of \(z\) or \(x, y\) where \(z = x+iy\).
Rearranging a bit will get you that \(w=\frac{1}{3} \frac{z+4i}{(z-1)} = \frac{1}{3} \frac{x+i(y+4)}{(x-1)+iy}\) - realising the denominator here will give you some expression for \(w\) such that you can split it into its real and imaginary parts. After doing this, it should be trivial enough to obtain an expression for \(w\) such that \(w\) is purely imaginary ie. \(\text{Re}(w) = 0\).

Hope this helps :)
Spoiler
HSC 2018: Mod Hist [88] | 2U Maths [98]
HSC 2019: Physics [92] | Chemistry [93] | English Adv [87] | 3U Maths [98] | 4U Maths [97]
ATAR: 99.05

UCAT: 3310 - VR [740] | DM [890] | QR [880] | AR [800]
Guide Links:
Subject Acceleration (2018)
UCAT Question Compilation/FAQ (2020)
Asking good questions

nat_21

  • Fresh Poster
  • *
  • Posts: 3
  • Respect: 0
Re: Help with Complex number Loci questions
« Reply #4 on: March 06, 2021, 05:05:31 pm »
+1
I think from the outset your method is wrong - you're assuming that \(w\) is purely imaginary then finding the locus some \(z\) such that \(z=x+iy\). It should be done in the opposite way ie. you choose some \(z\) such that \(z=x+iy\) and hence determine the conditions on \(x, y\) (ie. the locus!) such that \(w\) is imaginary.

Realising the denominator immediately doesn't help too much - we ideally want to express w in terms of \(z\) or \(x, y\) where \(z = x+iy\).
Rearranging a bit will get you that \(w=\frac{1}{3} \frac{z+4i}{(z-1)} = \frac{1}{3} \frac{x+i(y+4)}{(x-1)+iy}\) - realising the denominator here will give you some expression for \(w\) such that you can split it into its real and imaginary parts. After doing this, it should be trivial enough to obtain an expression for \(w\) such that \(w\) is purely imaginary ie. \(\text{Re}(w) = 0\).

Hope this helps :)
It works! I found the answer, thank you very much!!  :)