Problem:
Let z1=18+83i,z2=18+39i, and z3=78+99i, where i=−1. Let z be the unique complex number with the properties that z2−z1z3−z1⋅z−z3z−z2 is a real number and the imaginary part of z is the greatest possible. Find the real part of z.
Solution:
Let z2−z1z3−z1=r1cis(θ1), where 0∘<θ1<180∘.
If z is on or below the line through z2 and z3, then z−z3z−z2=r2cis(θ2), where 0∘<θ2<180∘. Because r1cis(θ1)⋅r2cis(θ2)=r1⋅r2⋅cis(θ1+θ2) is real, it follows that θ1+θ2=180∘, meaning that z1,z2,z3, and z lie on a circle. On the other hand, if z is above the line through z2 and z3, then z−z3z−z2=r2cis(−θ2), where 0∘<θ2<180∘. Because r1cis(θ1)⋅r2cis(θ2)=r1⋅r2⋅cis(θ1−θ2) is real, it follows that θ1=θ2, meaning that z1,z2,z3, and z lie on a circle. In either case, z must lie on the circumcircle of △z1z2z3, whose center is the intersection of the perpendicular bisectors of z1z2 and z1z3, namely, the lines y=239+83=61 and 16(y−91)=−60(x−48). Thus the center of the circle is c=56+61i. The imaginary part of z is maximal when z is at the top of the circle, and the real part of z is 56.
OR
Let z=a+bi, where a and b are real numbers. Then the given expression is
which simplifies to (a−56)2+(b−61)2=1928. Thus a+bi lies on the circle centered at 56+61i with radius 1928. When b is maximal, z is at the top of the circle, and a=56.
