Question

If $|z-4+3i|\le 2,$ then the least and the greatest values of $\left|z\right|$ are

• A. $3,7$
• B. $4,7$
• C. $3,9$
• D. None of these

Answer 1

The correct option is A $3,7$

Let $z=x+iy$

Given, $|z-4+3i|\le 2.$

$\therefore |x+iy-4+3i|\le 2.$

or, $\sqrt{{(x-4)}^2+{(y+3)}^2}\le 2$

or, ${(x-4)}^2+{(y+3)}^2\le 2^2$

Thus, $z$ lies in the interior or on the boundary of the circle whose centre is $(4,-3)$ and radius is $2.$

Least value of $\left|z\right|=OA=OC-AC=5-2=3.$

Greatest value of $\left|z\right|=OB=OC+CB=5+2=7.$

Thus, $3\le \left|z\right|\le 7.$