# Complex number maximum and minimum values.

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.

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.

• Thanks for geometric approach. But why my algebraic approach is giving a wrong answer? Oct 6, 2016 at 9:22
• I guess you correctly proved that the module is $\le 6 + \sqrt(34)$. This is correct, but that does not mean that it CAN reach this value for some $z$. Oct 6, 2016 at 9:47
• When calculating the minimum I guess you have a mistake in: $$|\iota z+z_1| \ge |\iota z|_{min}+ |z_1|$$ For z for which the module reaches minimum shouldn't it be $$|\iota z+z_1| \le |\iota z|+ |z_1|$$? Oct 6, 2016 at 9:55
• Sir as you mentioned that it is not necessary that the function can reach this value, then should not I use algebraic methods to find the maximum and minimum values of the function? Oct 6, 2016 at 11:06
• May be it's possible to use algebraic approaches to this problem, but I just do not know how it can be done. One more illustration, why your approach is not good. Consider more simple problem: what is the possible maximum if $|1 + \iota|$? (of course it can be only $\sqrt{2}$, so this is the maximum). With your approach you can prove that $|1 + \iota| \le |1| + |\iota| = 2$. This is correct, but that does not mean that 2 is the maximum. Maximum is $\sqrt{2}$. Oct 6, 2016 at 11:44

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

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.