If z is a complex number such that |z|=3, then find the maximum value of the expression |z-i|/|z+1|.
Given x^2+y^2=9
maximize sqrt(x^2+(y-1)^2)/sqrt((x+1)^2+y^2)
letting y= +/-sqrt(9-x^2)
and substituting into the expression, observe by the graph that the maximum will occur with the minus sign.
This is g(x) on the Desmos graph.
https://www.desmos.com/calculator/yt7vcft65j
Line 4 is the derivative. Line 5 is solving the numerator = 0.
The solution for x, called a is on line 7.
line 8 gives the answer to the problem g(a)=1.66225322816
The following lines are just the closed form, trying to simplify things.
The final line may be the nicest, but it still has a triple-nested square root.
\frac{1}{4}\sqrt{\frac{41+3\sqrt{41}}{82}\left(50+\sqrt{450-54\sqrt{41}}\right)}
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Posted by Jer
on 2024-06-14 15:31:30 |
exact solution
