Complex number maximum and minimum values.

Question

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.

Answer 1

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.

Answer 2

When $\text{z}\in\mathbb{C}$: $$\text{z}=\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i$$

So, we get:

1. $$\left|\text{z}-i\right|\le5\Longleftrightarrow\left|\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i-i\right|\le5\Longleftrightarrow\sqrt{\Re^2\left[\text{z}\right]+\left(\Im\left[\text{z}\right]-1\right)^2}\le5$$
2. $$ \begin{cases} \left|\text{z}i+\text{z}_1\right|\\ \text{z}_1=5+3i \end{cases}\Longleftrightarrow \begin{cases} \left|\left(\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i\right)i+5+3i\right|\\ \text{z}_1=5+3i \end{cases}\therefore\sqrt{\left(5-\Im\left[\text{z}\right]\right)^2+\left(3+\Re\left[\text{z}\right]\right)^2} $$

Now, for:

$$\sqrt{\Re^2\left[\text{z}\right]+\left(\Im\left[\text{z}\right]-1\right)^2}\le5$$

We find:

1. For $\Re\left[\text{z}\right]$: $$-5<\Re\left[\text{z}\right]<5$$
2. For $\Im\left[\text{z}\right]$: $$1-\sqrt{25-\Re^2\left[\text{z}\right]}\le\Im\left[\text{z}\right]\le1+\sqrt{25-\Re^2\left[\text{z}\right]}$$

Answer 3

Because the right equality of triangle inequality holds when two complex numbers are on one line and they are on same side. Since the formula of this complex express an equation of circle obviously, you need to calculate the distance to center of circle. The opposite of circle is max, near side of circle from origin is minimum, generally. Therefore the answer is as lesnik say.