Complex Geometry (1 Viewer)

[Aysce]

Question: (Note I've done the first part)

10.a. Prove that the points on an Argand diagram which represent the complex numbers and where k is a real number, form the vertices of a right angled triangle. You may assume the right angle is at the point .

b. For what values of k is the above triangle an isosceles right-angled triangle?


This question is from Terry Lee and do they happen to ask these sort of questions in HSC or trial papers?

 

[largarithmic]

Question: (Note I've done the first part)

10.a. Prove that the points on an Argand diagram which represent the complex numbers and where k is a real number, form the vertices of a right angled triangle. You may assume the right angle is at the point .

b. For what values of k is the above triangle an isosceles right-angled triangle?


This question is from Terry Lee and do they happen to ask these sort of questions in HSC or trial papers?

Let A represent z1, B represent z2, C represent the other thing.

Consider this:





This number is clearly imaginary, i.e. the complex number CA divided by CB is purely imaginary, i.e. CA and CB are at right anlges to eachother. The triangle formed is then isosceles if |CA|=|CB| or if |CA/CB| = 1: which is if or just k = 1 or -1.

 

[Aysce]

Umm I don't really get how you got l 1/-k l = 1 ?

 

[largarithmic]

Umm I don't really get how you got l 1/-k l = 1 ?

Alright, that quantity i/-k represents the vector CA divided by the vector CB, right? Now if A,B,C is isosceles right angled at C, that means that the length CA = the length CB, right? I.e. |CA| = |CB|, i.e. |CA/CB| = 1 right? So you have to get |i/-k| = 1, coz CA/CB = i/-k if CA and CB are written in complex number form. But i has unit modulus so you can sorta ignore it in |i/-k| = 1 and just get |i/-k| = 1 or just |k| = 1.

 

[Aysce]

Oh I get it now, thanks a lot man I'm having lots of trouble with geometry in complex numbers

 

[SpiralFlex]

I think I have seen this question in a past paper once. Though it was just a variation of the one from Terry Lee.