Problem:
For complex numbers u=a+bi and v=c+di (where i=−1 ), define the binary operation
u⊗v=ac+bdi
Suppose z is a complex number such that z⊗z=z2+40. What is ∣z∣?
Answer Choices:
A. 2
B. 5
C. 5
D. 10
E. 52
Solution:
Let z=a+bi. Then z⊗z=a2+b2i and z2=(a2−b2)+2abi, so the original equation is equivalent to
a2+b2i=(a2−b2+40)+2abi
Setting the real parts of both sides equal gives b2=40, so b=±210. Setting the imaginary parts of both sides equal yields a=2b=±10. Then
∣z∣=a2+b2=10+40=(E)52
The problems on this page are the property of the MAA's American Mathematics Competitions