Question

If $a,b,c$ are real , then $az\overline{z}+bz+\overline{b}z+c=0$ represents

• A. circle with centre $\frac{b}{a}$ and radius $\frac{1}{a}\sqrt{|b{|}^2-ca}$
• B. circle with centre $-\frac{b}{a}$ and radius
  $\frac{1}{a}\sqrt{|b{|}^2-ca}$
• C. Circle with centre $\frac{b}{a}$ and radius $\sqrt{|b{|}^2-ca}$
• D. circle with centre $-\frac{b}{a}$ and radius $\sqrt{|b{|}^2-ca}$

Answer 1

Answer: (B)

circle with centre $-\frac{b}{a}$ and radius

$\frac{1}{a}\sqrt{|b{|}^2-ca}$
let z=x+iy, $\overline{z}=x-iy$
$\therefore az\overline{z}+bz+b\overline{z}+c=0$
$\Rightarrow a|z{|}^2+bz+b\overline{z}+c=0$
$\Rightarrow a(x^2+y^2)+b(x+iy)+b(x-iy)+c=0$
$\Rightarrow a(x^2+y^2+2bx+c=0)$
$x^2+y^2+\frac{2b}{a}x+0y+\frac{c}{a}=0$
comparing with general equation of circle,
$x^2+y^2+2gx+2by+c=0,$
centre $(-g,-f)\Rightarrow (-b/a,0)$
radius$=\sqrt{g^2+f^2-c}$
$=\sqrt{b^2/a^2-c/a}$
$=\left(\sqrt{b^2-ac}\right)/a$