Complex help (1 Viewer)

[cutemouse]

Hi, could someone please help me with the following complex numbers question? Thanks

Let l be the line in the complex plane that passes through the orgin and makes an angle α with the positive real axis, where 0 < α < pi/2

(SEE ATTACHMENT)

The point P represents the complex number z<sub>1[/sup], where 0<arg(z1)<α. The point P is reflected in the line l to produce the point Q, which presents the complex number z2. Hence |z1=|z2|

i) Explain why arg(z1)+arg(z2)=2α

ii) Deduce that z1z2=|z1|²(cos2α + isin2α)

iii) Let α=pi/4 and let R be the point that represents the complex number z1z2. Describe the locus of R as z1 varies.

Thanks!</sub>

 

[conics2008]

Hey There

i) see that diagram, ill show u the common sense way..

Let arg z1 = theta therefore angle POL = alpha - theta

then construct a triangle POQ since |z1|=|z2| its an isoscles.

therefore arg z2 = alpha- theta +alpah = 2alpha - theta

therefore arg z1 + arg z2 = theta + 2alpha - theta = 2alpah =]

there are alot of methods, this is the way i see it.

ii) we know z1 = |z1|(cos theta + isin theta) = |z1|cis(theta)
and z2= |z2|(cos 2alpha-theta +i sin 2alpha-theta ) but we know that |z1|=|z2|

there fore z1.z2 = |z1|cis(theta) * |z1| cis (2alpha - theta)

there fore we add arguments because its multiply...

there fore |z1|^2 cis (2alpha) = z1*z2=|z1|^2(cos2a+isin2a)

iii) just sub in pi/4 into z1*z2 and know wtf it is...

 

[cutemouse]

iii) just sub in pi/4 into z1*z2 and know wtf it is...

Hi,

Firstly thanks for your help... But I get |z₁|².i when I sub in pi/4... So what type of locus would that be?

Thanks again