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The points representing the complex number z for which arg $\left(\frac{z-2}{z+2}\right)=\frac{\pi}{3}$ lie on
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Verified Answer
The correct answer is:
a circle
Let $z=x+i y$ $\therefore \quad \frac{z-2}{z+2}=\frac{x+iy-2}{x+i y+2}$
$=\frac{(x-2)+i y}{(x+2)+6 y} \times \frac{(x+2)-i y}{(x+2)-i y}$
$=\frac{(x-2)(x+2)+i y(x+2)-i y(x-2)-i^{2} y^{2}}{(x+2)^{2}-(i y)^{2}}$
$=\frac{x^{2}-4+i x y+2 i y-i x y+2 i y+y^{2}}{(x+2)^{2}+y^{2}}$
$x^{2}+y^{2}-4-\frac{4}{\sqrt{3}} y=0,$ which represents a
circle.
$=\frac{(x-2)+i y}{(x+2)+6 y} \times \frac{(x+2)-i y}{(x+2)-i y}$
$=\frac{(x-2)(x+2)+i y(x+2)-i y(x-2)-i^{2} y^{2}}{(x+2)^{2}-(i y)^{2}}$
$=\frac{x^{2}-4+i x y+2 i y-i x y+2 i y+y^{2}}{(x+2)^{2}+y^{2}}$
$x^{2}+y^{2}-4-\frac{4}{\sqrt{3}} y=0,$ which represents a
circle.
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