If $|z-\iota|\le5$ and $z_1=5+3\iota$ where $\iota= \sqrt{-1}$, then what would be the greatest and least value of $|\iota z+z_1|$ ?
My Attempt:
Now I know that if there are two complex numbers namely $a$ and $b$ then
$$||a|-|b|| \le |a+b| \le |a|+|b| $$
Going according to this $$|z+(-\iota)| \ge ||z|-|(-\iota)||$$
$$|z-\iota| \ge ||z|-1| \tag1$$
From equation $(1)$ and using the input from questions (i.e. $|z-\iota|\le5$)
$$\Rightarrow ||z|-1| \le 5$$
$$\Rightarrow -5 \le |z|-1 \le 5$$
$$\Rightarrow -4 \le |z| \le 6$$
$$\Rightarrow -4|\iota| \le |z||\iota| \le 6|\iota|$$
$$\Rightarrow -4 \le |z\iota| \le 6$$
Now its easy to deduce that $$|z_1|=\sqrt{5^2+3^2}$$
Now the greatest value of$ |\iota z+z_1|$ would be
$$|\iota z+z_1| \le |\iota z|_{max}+ |z_1|$$
$$\Rightarrow |\iota z+z_1| \le 6+ \sqrt{34}$$
Similarly the least value of$ |\iota z+z_1|$ would be
$$|\iota z+z_1| \ge |\iota z|_{min}- |z_1|$$
$$|\iota z+z_1| \ge -\sqrt{34}-4$$
But my book says that the maximum value should $10$ and the minimum value must be $0$. Why is my answer wrong? ANY KIND OF HINT WOULD WORK.
Best Answer
I would suggest geometric approach.
$z$ such that $|z-\iota|\le5$ are all located inside a circle with center at $\iota$ and with radius 5.
Now multiply all these $z$'s by $\iota$. The whole picture turns around by $\pi /4$ and we get a circle of radius 5 with center at -1.
Now we add $5 + 3 \iota$. The whole circle moves and it's center is now at $4 + 3 \iota$. Distance from the $0$ to the center is exactly 5. So, the minimum distance from $0$ to some point on circle is 0, the maximum is 10.