Question

Number of complex numbers satisfying equation $z^3=\overline{z}$ & arg $(z+1)=\frac{\pi }{4}$ simultaneously is-

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

Let $z=x+iy$

Applying given conditions ;;

$\Rightarrow (x+iy{)}^3=x-iy$

$\Rightarrow x^3-3x{y}^2-i\left(y^{3}g3x^{2}y\right)=x-iy$

Comparing both sides;

$\Rightarrow x^3-3x{y}^2=x$ and $y^3-3x^{2}y=y$

Applying $arg(z+1)=\frac{\pi }{4}$

$\Rightarrow \frac{y}{x+1}=1\Rightarrow y=x+1$

Solving first two conditions ;

$\Rightarrow x^2-3y^2=y^2-3x^2$

$\Rightarrow 4x^2=4y^2$

$\Rightarrow x=\pm y$

Substituting above in 3rd equation ;

$\Rightarrow y=-y+1$

$\Rightarrow y=\frac{1}{2}$

$\Rightarrow x=-\frac{1}{2}$

Therefore, only complex number satisfying the given condition is $\frac{1}{2}+i\frac{1}{2}$

Number of complex numbers satisfying given conditions simultaneously is 1