Newbie Complex Number Question (1 Viewer)

[Estel]

arg(z-1) - arg(z+1) = pi/4
prove locus of z is a fixed circle with centre at the pt representing i.

Please show a geometrical solution...

 

[CrashOveride]

err dont u mean major arc of a circle? thats how these questions turn out, u gotta draw the vectors going from z to 1 and z to -1 and using the data u can show that the apex angle of the triangle is pi/4. So the locus is the major arc of a circle and by observing symmetry, center lies on the y axis. call -1,0 X and 1,0 Y now u now XZA is 45 so XCA is 90. from that u shud be able to deduce that OC = OX=OY so the centre is at (0,1) or i.

 

[withoutaface]

arg([z-1]/[z+1])=pi/4

then you simplify using z=a+ib and say that overall if (z-1)/(z+1)=x+iy

then x=y and x>0 because this is the only way an arg of pi/4 can exist (ie if tan(y/x)=pi/4)

 

[Estel]

Thankyou very much!

Damn I'm finding complex numbers a million times harder than 3U... =/

 

[withoutaface]

Thankyou very much!

Damn I'm finding complex numbers a million times harder than 3U... =/

Once you've got it it stays with you forever, so just work at it.

 

[McLake]

Thankyou very much!

Damn I'm finding complex numbers a million times harder than 3U... =/

They are daunting at first, but soon they will become as easy to work with as surds or fractions ...

 

[Estel]

But I'm finding these questions hell lol...
Another:
Z₁ and Z₂ represent z₁ and z₂ respectively on the Argand Diagram. If together with the origin, the two points form an isosceles triangle right angled at O, prove z₁² + z₂² = 0

 

[withoutaface]

if it is a right angled trinangle then argz₁=argz₂+/-pi/2

hence argz₁²=2argz₂+pi

while argz₂²=2argz₂

hence one has an arg which is directly opposite to the other, and since it is isoceles then the mods are equal, hence the addition of the vectors equates to nothing.

 

[Estel]

Thankyou again...
how long did you spend on complex numbers?

 

[withoutaface]

can't remember, somewhere between 3-4 weeks, i remember we got in both complex and graphing after the 2u hsc before end of term (about 8th december)