Let a= u+ iv, b= p+ iq, z= x+ iy. Then [tex]\overline{a}z+ a\overline{z}+ b= 0[tex] becomes (u- iv)(x+ iy)+ (u+ iv)(x- iy)+ p+ iq= 0

(ux+ vy)+ i(uy- vx)+ (ux+ vy)+ i(vx- uy)+ p+ iq= 0

Notice that i(uy- vx) and i(vx-uy) cancel. Separating real and imaginary parts, 2ux+ 2vy+ p= 0 and q= 0 (the latter is why b= p is a real number).

We can rewrite the first equation as 2vy= -2ux+ b or y= -(u/v)x+ b/(2v) so that is, in fact, a straight line with slope -u/v and y-intercept b/(2v).

Now, I honestly don't see what that has to do with the "complex slope", ! Where did you see that formula?