Question

If $\overline{a}=b,c\in R$ and $|b{|}^2>c$, then $z\overline{z}+az+b\overline{z}+c=0$ represents

• A. A circle
• B. A parabola
• C. A straight line
• D. Finite number of points

Answer 1

Answer: (A)

A circle

$z\overline{z}+az+b\overline{z}+c=0$
Or
$|z{|}^2+az+\overline{a}\overline{z}+c=0$
Or
$az+\overline{az}+c+|z{|}^2=0$
If $z=x+iy$ we get and $a=a_x+i{a}_y$
$x^2+y^2+(a_x+i{a}_y)(x+iy)+(a_x-i{a}_y)(x-iy)+c=0$
$x^2+y^2+x{a}_x-y{a}_x+i(x{a}_y+y{a}_x)+x{a}_x-y{a}_x-i(x{a}_y+y{a}_x)+c=0$
Or
$x^2+y^2+2x{a}_x-2y{a}_x+c=0$
This represents an equation of a circle.