I am having trouble with the following question. I am not asking domeone to explicitly solve it for me; just an indication of where I’m going wrong or an helpful insight is more than enough.
Find the set of points, $z$, for which $\displaystyle\arg\left(\frac{z-i}{z+i}\right)=\pi/4$
Although I understand the question geometrically (finding points $z$ such that the angle between the vectors $z-i$ & $z+i$ is $45°$). I Have no idea at all on solving this geometrically.
Solving algebraically, I conclude that $z$ must lie on a circle of radius $\sqrt 2$ centered at $(-1,0)$. But the answer given only includes the major arc subtended by the segment joining $i$ & $-i$.
Where am I going wrong?
Best Answer
When you write $$ \arg(x^2+y^2-1-2xi)=\frac{\pi}{4} $$ you don't know if the real part, $x^2+y^2-1,$ is or not positive, so you should set the condition $$ x^2+y^2-1>0\tag1 $$ together with the condition that you already found $$ x^2+y^2-1=-2x.\tag2 $$ The condition $(1),$ by using $(2)$ gives $-2x>0,$ or finally $x<0,$ so it excludes exactly the part of the circle required.