Question

help!!
complex numbers

Answer 1

show that the plot of this on the imaginary plane is a circle \(\omega\) is the only variable and varies from zero to infinity(only +ve values)
\[\frac{ (b \omega )i }{ (k-\omega^2) +(b \omega)i }\]

Answer 2

hey men i have not done them yet sorry if i did i would help u well if i have i dont member them

Answer 3

if you want the original context, skip to 8:05

https://www.youtube.com/watch?v=R_8beV_gXHc

Answer 4

thats ok @likeabossssssss :)

Answer 5

@ganeshie8

Answer 6

^if you saw the vid, he says that the circle is independent of b,k

i have tried writing the equation of a circle with shifted centre in the complex plane and comparing, i am getting nowhere

Answer 7

Usually when I hear about circle in complex numbers, I try to bring in modulus of z-something=constant.

Answer 8

yeah that works nicely, simply write the given complex number in standard form : \(x(t)+iy(t)\)

then show that \((x-a)^2+(y-b)^2=r^2\)

Answer 9

\[\frac{ (b \omega)^2 +(b \omega)(k- \omega^2)i}{(k- \omega^2)^2+(b \omega)^2 }\]

Answer 10

^ is that right?
what do i do next?

Answer 11

looks good, take the modulus and see if it is a constant

Answer 12

the modulus wont be a constant, distance of the point on the plot from the origin is varying

Answer 13

the vid also says, radius = 1/2 and centre: x=1/2

Answer 14

@ganeshie8 can we apply limits to the "argument" of the complex number and see what happens when \(\omega\) tends to infinity?

i dont think its circular like \(\theta\), the plot will only complete one circle as \(\omega \) tends to infinity

Answer 15

Ohk..

Answer 16

^i think, I dont know...

Answer 17

http://www.wolframalpha.com/input/?i=%28%28b*w%29%5E2%2F%28%28k-w%5E2%29%5E2%2B%28b*w%29%5E2%29+-+1%2F2%29%5E2+%2B+%28%28b*w%29%28k-w%5E2%29%2F%28%28k-w%5E2%29%5E2%2B%28b*w%29%5E2%29+%29%5E2

Answer 18

that does show (x - 1/2)^2 + y^2 = 1/4

Answer 19

thanks!!
but do you see how we can work backwards and find center, radius?
because i have other cases, where the center, radius does depend on b,k..

the material doesn't ask me to show that the plot is a circle...but i feel i'm missing out on something very obvious

Answer 20

drove me slightly crazy but here it goes:

first to simplify the eqns, let A=k-w^2 and B=bw

so the complex no becomes Bi/(A+Bi) which becomes
\[\frac{ B^2 }{ A^2 +B^2 } + \frac{ AB }{ A^2 +B^2 }i\]
nice-looking couple arent they? ;)

converting to polar,
\[\tan(\theta)=A/B\] and
\[r^2 = \frac{ B^4 + A^2B^2 }{(A^2+B^2)^2 }\]
substituting \[A=B\tan(\theta)\] into the above
\[r^2=\frac{ B^4(1+\tan^2(\theta)) }{B^4(\tan^2(\theta)+1)^2 }\]
\[=\frac{ 1}{\tan^2(\theta)+1}=\frac{ 1}{ \sec^2(\theta) }=\cos^2(\theta)\]

\[r=cos(\theta)\]
which is a circle center at (1/2,0) n radius 1/2

enjoy :)

Answer 21

excellent approach @superdavesuper
but i still do not follow how you concluded that its a circle with C(1/2,0) and r=1/2
can you explain?

Answer 22

@baru its a standard form for circle in polar coordinates; see this:

http://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx

Answer 23

@superdavesuper thanks!! i'll have a look when i can

Answer 24

Nice!