Halp complex locus (1 Viewer)

SammyT123 · Mar 26, 2016

Find the locus of w such that w=(z-2)/z and that |z|=1
How do I approach this question , I tried to split the fraction and multiply the numerator/denominator of 2/z by the conjugate of z
Didn't work , incorrect answer

Thankyou

integral95 · Mar 26, 2016

Make z the subject of the equation and take the moduli of both sides.

You should get

$|w-1| = 2$

Which is a circle centred at (1,0) with radius 2

KingOfActing · Mar 26, 2016

SammyT123 said:
Find the locus of w such that w=(z-2)/z and that |z|=1
How do I approach this question , I tried to split the fraction and multiply the numerator/denominator of 2/z by the conjugate of z
Didn't work , incorrect answer

Thankyou

Haven't done these in a bit but I reckon this is fine:

$$Let $ w = x + iy$ and $ z = \mathrm{cis}(\theta) $ (note that we can do this since $|z| = 1$.$\\x + iy = \frac{z-2}{z} = 1 - \frac{2}{ \mathrm{cis}(\theta)}\\x + iy = 1 - 2 \mathrm{cis}(-\theta) = 1 - 2\cos{\theta} + 2i\sin{\theta}\\$Equating real and imaginary parts: $\\x = 1 - 2\cos{\theta}$, $ y = 2\sin{\theta}\\(x-1)^2 + y^2 = 4$

SammyT123 · Mar 26, 2016

integral95 said:
Make z the subject of the equation and take the moduli of both sides.

You should get

$|w-1| = 2$

Which is a circle centred at (1,0) with radius 2

Thankyou so much

SammyT123 · Mar 26, 2016

KingOfActing said:
Haven't done these in a bit but I reckon this is fine:

$$Let $ w = x + iy$ and $ z = \mathrm{cis}(\theta) $ (note that we can do this since $|z| = 1$.$\\x + iy = \frac{z-2}{z} = 1 - \frac{2}{ \mathrm{cis}(\theta)}\\x + iy = 1 - 2 \mathrm{cis}(-\theta) = 1 - 2\cos{\theta} + 2i\sin{\theta}\\$Equating real and imaginary parts: $\\x = 1 - 2\cos{\theta}$, $ y = 2\sin{\theta}\\(x-1)^2 + y^2 = 4$

Hey, never thought about it this way
Thanks for helping

KingOfActing · Mar 26, 2016

SammyT123 said:
Hey, never thought about it this way
Thanks for helping

You're welcome

Halp complex locus (1 Viewer)

SammyT123

Active Member

integral95

Well-Known Member

KingOfActing

lukewarm mess

SammyT123

Active Member

SammyT123

Active Member

KingOfActing

lukewarm mess

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