If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$

Question

Question :

If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$

What I did :

$$|z-3i| \leq |z|+3 \tag i$$

Also $$|z-4| \leq |z| +4 \tag{ii}$$

Now adding (i) and (ii) we get

$$ \Rightarrow |z-3i|+|z-4| \leq 2|z| +7 $$

$$\Rightarrow 2|z| +7 \leq 5$$

$$\Rightarrow |z| \leq -1 $$

Please suggest whether this is correct or wrong , also suggest the mistake thanks.

Answer 1

$$\Rightarrow |z-3i|+|z-4| \leq 2|z| +7 \\$$ $$\Rightarrow 2|z| +7 \leq 5$$

Here is the mistake. Plugging the value, you would get $5\leq 2|z|+7$ which would ultimately give you $ |z| \geq -1$ which is is kind of useless since $|z| \geq 0$

Hint: Notice that the distance between $(0,3i)$ and $(4,0)$ equals $5$. If $z$ has to satisfy $|z-3i|+|z-4|=5$, then it has to lie on the segment joining $(0,3i)$ and $(4,0)$. When $z$ varies on the aforesaid line, its distance from origin (i.e, $|z|$) will be minimum if $z$ is the foot of perpendicular drawn from origin on the line segment.

Answer 2

Some hints:

Draw a figure! As $3^2+4^2=5^2$ the set $S$ of $z$ satisfying the given condition has a very simple description. It is then easy to see and compute the point $p\in S$ having smallest distance from the origin.

Answer 3

Note that the distance between the points $0+3i$ and $4+0i$ is $\sqrt{4^2 + 3^2} = 5$, so that any point satisfying the given condition $$f(z) = |z-3i| + |z-4| = 5$$ must lie on the line segment joining these two points. We can parametrize this as $$z(t) = 3i(1-t) + 4t, \quad 0 \le t \le 1.$$ Then the point that minimizes $|z(t)|$ can be found by taking $$|z(t)|^2 = 9-18t+25t^2$$ and differentiating with respect to $t$: $$\frac{d|z|^2}{dt} = 50t-18.$$ Thus the squared distance is minimized for $t = \frac{9}{25}$, corresponding to the value $z(9/25) = \frac{36+48i}{25}$, with minimum distance $|z| = 12/5$.

A somewhat less computational way to get the minimum distance from $|z(t)|^2$ is to complete the square: $$|z(t)|^2 = (5t-\tfrac{9}{5})^2 + \tfrac{144}{25},$$ and now it is immediately obvious that $|z(t)|$ has minimum value $12/5$ corresponding to the choice $t = 9/25 \in [0,1]$.