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.
My answer: 12.1875 pi = approx 38.2881604656256
For reference, the area between circles (in the real xy plane) of radius 2 and 4 is 12 pi.
Let z = a+bi
1/z = (a-bi)/(a^2+b^2)
z+1/z = a * (1 + 1/(a^2+b^2)) +
b * (1 - 1/(a^2+b^2))i
z+1/z = a * ((a^2+b^2+1)/(a^2+b^2)) +
b * ((a^2+b^2-1)/(a^2+b^2))i
The reciprocal of a complex number z has magnitude which is the reciprocal of |z| and the opposite angle.
How to apply that transformation to the 2 equations:
x^2 + y^2 = 16 and
x^2 + y^2 = 4
Replace x with: x * ((x^2+y^2+1)/(x^2+y^2))
Replace y with: y * ((x^2+y^2-1)/(x^2+y^2))
for Larger circle, ((x^2+y^2+1)/(x^2+y^2)) = 17/16
for Larger circle, ((x^2+y^2-1)/(x^2+y^2)) = 15/16
for Smaller circle, ((x^2+y^2+1)/(x^2+y^2)) = 5/4
for Smaller circle, ((x^2+y^2-1)/(x^2+y^2)) = 3/4
x^2 + y^2 = 16 --> (17/16)^2x^2 + (15/16)^2y^2 = 16
x^2 + y^2 = 4 --> (5/4)^2x^2 + (3/4)^2y^2 = 4
Prelim check with Desmos suggested an error. I had my ellipses morphed the wrong way; divide instead of multiply.
Area of ellipse is pi AB where A and B are the axes measured from the center.
So the larger ellipse Area is pi*(17/16)*(15/16)*16
And smaller ellipse Area is pi*(5/4)*(3/4)*4
Area = (15.9375 - 3.75) pi
Area = 12.1875 pi = approx 38.2881604656256
https://www.desmos.com/calculator/e4ftmo5buy
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Posted by Larry
on 2024-12-13 11:10:26 |
Algebraic Solution
