Question

If $z=x+iy,x,yϵR$ then locus of equation $b\overline{z}+z\overline{b}=c$ where $b,cϵR,b\ne 0$ are fixed,is

• A. a straight line
• B. an ellipse
• C. a circle
• D. a hyperbola

Answer 1

Answer: (A)

let
$b=m+ni$... fixed point.
$z=x+iy$ .... variable.
Hence
$\overline{z}b+\overline{b}z=c$
$(x-iy)(m+ni)+(m-ni)(x+iy)=c$
$mx+inx-imy+ny+mx+imy-inx+ny=c$
$2mx+2ny=c$
$x\left(2m\right)+y\left(2n\right)=c$
This is nothing but equation of a straight line.
Hence $z$ lies on a straight line.