Question

Let a=5/13 + 12i/13 and b=2-3i. The function h(z)=a*conj(z)+b$ represents a reflection in the complex plane. What is the line that h(z) reflects over?

Answer 1

the midpoint ov the line joining a and b is (31/26 , -27/26)
The slope of the line through a and b is -17/3
so the line of reflection is y + 27/16 = 3/17 (x - 31/26)

Also, if z = x+iy, then
h(z) = (5/13 x + 12/13 y + 2) + (12/13 x - 5/13 y - 3)i

See what you can do with that.