<s>[INST] Complex Numbers (Locus) Find the Cartesian equation of $z$ described by $$\arg\left(\frac{z-2}{z+5}\right)=\frac{\pi}{4}$$
So what I have done is let $z = x+iy$
$$\frac{z-2}{z+5} $$
$$ \frac{x+iy-2}{x+iy+5} $$
$$ \frac{x-2+iy}{x+5+iy} \frac{x+5-iy}{x+5-iy} $$(rationalizing here)
$$ \frac{(x-2)(x+5)+y^2+iy(x+5)-iy(x-2)}{(x+5)^2+y^2}$$
As $\arg\left(\frac{z-2}{z+5}\right)=\frac{\pi}{4}$
$$\arctan(\frac{Im(z)}{Re(z)}) = \frac{\pi}{4}$$
$$\frac{Im(z)}{Re(z)} = \tan(\frac{\pi}{4}) = 1$$
Therefore $$\frac{Im(z)}{Re(z)} = 1$$
Now comparing $im(z)$ and $re(z)$
$$\frac{y(x+5) - y(x-2)}{(x-2)(x+5)+y^2} = 1$$
$$7y = x^2 + 5x - 2x - 10 + y^2$$
$$x^2 + 3x - 10 + y^2 - 7y = 0$$
Now completing the square
$$(x+1.5)^2 + (y - 3.5)^2 = 14.5$$
Therefore locus is a circle with a centre of $(-1.5$ and $3.5i)$ with radius of $\sqrt{14.5}$
Is this correct?
 [/INST] Yes ! It is very easy if you can interpret geometrically . See the red lines in the picture .

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