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(C) Let $z = x + iy$. Then $\frac{z - 3i}{z + 4} = \frac{x + i(y - 3)}{(x + 4) + iy}$. Multiplying numerator and denominator by the conjugate of the d

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$x^2 + y^2 + 4 x - 3 y = 0$ and $3 x - 4 y < 0$

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(C) Let $z = x + iy$. Then $\frac{z - 3i}{z + 4} = \frac{x + i(y - 3)}{(x + 4) + iy}$. Multiplying numerator and denominator by the conjugate of the denominator: $\frac{x + i(y - 3)}{(x + 4) + iy} imes \frac{(x + 4) - iy}{(x + 4) - iy} = \frac{x(x + 4) - xyi + i(y - 3)(x + 4) + y(y - 3)}{(x + 4)^2 + y^2}$. The real part is $\frac{x^2 + 4x + y^2 - 3y}{(x + 4)^2 + y^2}$ and the imaginary part is $\frac{xy - 3x + 4y - 12 - xy + 4x}{(x + 4)^2 + y^2} = \frac{4y - 3x - 12}{(x + 4)^2 + y^2}$. Since $\operatorname{Arg}(w) = \frac{\pi}{2}$,the real part must be $0$ and the imaginary part must be positive. Thus,$x^2 + y^2 + 4x - 3y = 0$ and $4y - 3x - 12 > 0$. Note that the expression simplifies to $\frac{x^2 + y^2 + 4x - 3y + i(4y - 3x - 12)}{(x + 4)^2 + y^2}$. Setting the real part to $0$ gives $x^2 + y^2 + 4x - 3y = 0$. The condition for $\operatorname{Arg} = \frac{\pi}{2}$ is $ ext{Re}(w) = 0$ and $ ext{Im}(w) > 0$. Therefore,$4y - 3x - 12 > 0$,which implies $3x - 4y < -12 < 0$.

If $P(x, y)$ 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

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