If $\mathrm{P}(x, y)$ represents the complex number $z=x+i y$ in the Argand plane and…

If $\mathrm{P}(x, y)$ represents the complex number $z=x+i y$ in the Argand plane and $\operatorname{Arg}\left(\frac{z-3 i}{z+4}\right)=\frac{\pi}{2}$, then the equation of the locus of P is
  1. $x^2+y^2+4 x-3 y=0$ and $3 x-4 y\gt0$
  2. $x^2+y^2+4 x-3 y+2=0$ and $3 x-4 y\gt0$
  3. $x^2+y^2+4 x-3 y=0$ and $3 x-4 y \lt 0$
  4. $x^2+y^2+4 x-3 y+2=0$ and $3 x-4 y \lt 0$

Solution

$\begin{aligned} & \frac{z-3 i}{z+4}=\frac{x+(y-3) i}{(x+4)+y i} \times \frac{(x+4)-y i}{(x+4)-y i} \\ &= \frac{x(x+4)-x y i+(x+4)(y-3) i+y(y-3)}{(x+4)^2+y^2} \\ &= \frac{x^2+y^2+4 x-3 y+(4 x-3 x) i}{(x+4)^2+y^2} \\ & \operatorname{Arg}\left(\frac{z-3 i}{z+4}\right)=\frac{\pi}{2} \Rightarrow \tan \frac{\pi}{2}=\frac{4 y-3 x}{x^2+y^2+4 x-3 y} \\ & \Rightarrow x^2+y^2+4 x-3 y=0 \end{aligned}$
Also, for $\mathrm{I}^{\text {st }}$ quadrant, $4 y-3 x\gt0 \Rightarrow 3 x-4 y \lt 0$

Asked in: AP EAMCET 2024 (22 May Shift 2)

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