On an Argand diagram, sketch the loci of points representing complex numbers $w$ and $z$ such that $|w − 1 − 2i|= 1$ and $\arg(z-1)=\dfrac{3}{4}\pi$.
Find the least value of $|w-z|$ for points on these loci.
My attempt, I've already drawn the loci of w. But I don't know how to draw the arg one. Can anyone explain it to me how to draw and how to find the least value. Thanks a lot.
Best Answer
Basic approach. The locus of $\arg (z-1) = \frac{3\pi}{4}$ is that set of points $z$ for which a ray drawn from the point represented by $1$ on the complex plane, through $z$, makes an angle of $\frac{3\pi}{4}$ with the real axis.
That should also tell you which point on the locus of $w$ is the closest to the locus of $z$: Draw a line from $1+2i$, perpendicular to the ray representing the locus of $z$. Where that line intersects the loci of $w$ and $z$ are the desired points.