Let z be a complex number with 2≤|z|≤4. When all possible values of z+1/z are graphed on the complex plane, they form a region R. Compute the area of R.
In polar form z=r(cos(theta)+isin(theta))
and 1/z=1/r(cos(theta)-isin(theta))
z+1/z = (r+1/r)cos(theta) + (r-1/r)sin(theta)
This is an ellipse with major axis (r+1/r) and minor axis (1-1/r)
This is greatest when |z|=4 and has area pi*(17/4)(15/4) = 255pi/16
And smallest when |z|=2 and has area pi*(5/2)(3/2) = 15pi/4
The difference is 196pi/16 or about 38.29
Desmos added a complex mode and I finally got a chance to use it:
https://www.desmos.com/calculator/56wndswz4r
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Posted by Jer
on 2024-12-11 11:34:07 |
No Subject
