Question

If $|z-(4+4i\left)\right|\ge 4$, then area of the region bounded by the locii of $z,iz,-z$ and $-iz$ is:

• A. $4(4-\pi )$
• B. $16(4-\pi )$
• C. $16(\pi -1)$
• D. $4(\pi -1)$

Answer 1

The correct option is B $16(4-\pi )$

Locus of $|z-(4+4i)|=4$ is a circle with center at $(4,4)$ and radius $4$ is Complex plane.

Hence, locus of $|z-(4+4i)|\ge 4$ is all points either on or outside the circle with radius $4$ and center $(4,4)$.

Similarly, locus of $|-z-(4+4i)|\ge 4$ is all points on or outside the circle with radius $4$ and center $(-4,-4)$.

Locus of $|iz-(4+4i)|\ge 4$ is all points on or outside the circle with radius $4$ and center $(-4,4)$.

Finally, locus of $|-iz-(4+4i)|\ge 4$ is all points on or outside the circle with radius $4$ and center $(4,-4)$.

Hence, the area bounded by locus of all four will be the area enclosed by the four circles in argand plane as shown in the figure.
Area bounded$=$ area of shaded region
$=64-\pi r^2$
$=64-16\pi =16(4-\pi )$