Can we prove a arc as circle

Question

In this question ,according to me

The argument term shows a part of circle(major arc) and we have to prove it a circle , how it can be possible.

Answer 1

Let $z=x + iy $. We then get $$\arg (\frac {z-z_1}{z-z_2}) =\pi/4$$ $$\Rightarrow \arg (x-10+i (y-6)) -\arg (x-4) +i (y-6)) =\pi/4 $$ $$\Rightarrow \arctan (\frac {y-6}{x-10}) - \arctan (\frac {y-6}{x-4}) =\pi/4$$ Using the relevant formula, we have, $$ \frac {y-6}{x-10} -\frac {y-6}{x-4} = 1+ \frac {y-6}{x-10}\frac {y-6}{x-10}$$ $$\Rightarrow (x-7)^2 + (y-9)^2 = 18$$ $$\Rightarrow |z -7 -9i|^2 =18 $$ $$\boxed {|z-7-9i| = 3\sqrt{2}}$$ Hope it helps.