For complex numbers u=a+bi and v=c+di, define the binary operation by u v=ac+bdi. Suppose z is a complex number such that z z=z^2+40. What is |z|?

2023 AMC 12B Problem 12

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12.

For complex numbers

u=a+bi

and

v=c+di,

define the binary operation

\otimes

u\otimes v=ac+bdi.

Suppose

z

is a complex number such that

z\otimes z=z^2+40.

What is

|z|?

2

5

\sqrt{5}

\sqrt{10}

5\sqrt{2}

Write

z=a+bi;

then

z\otimes z=a^2+b^2 i

while

z^2=a^2-b^2+2abi

Match real and imaginary parts of

z\otimes z=z^2+40,

then compute

a^2+b^2

With

z=a+bi,

we have

z\otimes z=a^2+b^2 i

and

z^2+40=(a^2-b^2+40)+2abi.

The real parts give

a^2=a^2-b^2+40,

b^2=40.

The imaginary parts give

b^2=2ab,

b=2a

and

a^2=\tfrac{b^2}{4}=10.

Then

|z|^2=a^2+b^2=50,

|z|=5\sqrt{2}.

Thus, the correct answer is E.