how to find greatest and least values of $|z-2-3i|$ if $|z-5-7i|=9$ Context: conceptual question in jee
I got the answer by using properties of modulus but i don't know if the procedure is correct

If $|z-5-7i|=9$ then find the greatest and least values of $|z-2-3i|$
 Answer: $14$, $4$

here's what I did: 

given,
$$||z|- |-5-7i||\leq |z-5-7i| \leq |z| + |-5-7i|$$
$$||z|- \sqrt{74}|\leq 9 \leq |z| + \sqrt{74}$$
idk if what i'm about to do next is correct I did this
$$9\leq|z|+\sqrt{74}=> |z| \geq 9-\sqrt{74}$$
I assumed $|z| = 9-\sqrt{74}$
$$||z|-|-2-3i|| \leq|z-(2+3i)|\leq|z|+|-2-3i|$$
$$|9-\sqrt{74}-5| \leq |z-(2+3i) \leq 9+5$$
$$|-4.60| \leq |z-(2+3i) \leq 14$$
Again I assumed $|-4.60|$ approx $4$
$$4\leq|z-(2+3i) \leq 14$$
Lookig at the question $|z-(5+7i)| = 9$ resembles equation of circle whose center at $5+7i$with radius 9, is there any way to solve this question using circle equation? and how does equation of cirlce co-relate to finding greatest and least value of $|z-2-3i|$

Is what i did correct above cause I got that answer by randomly by plugging value of $|z|$ as I can't find solid solution to this question in any module/internet/book.

The set $c=\{z\in\Bbb C\mid|z-5-7i|=9\}$ is the circle centered at $C=5+7i$ with radius $9$. The points at which a point of that circle is nearest or further away from $P=2+3i$ are the points at which the line defined by $P$ and $C$ intersects the circle $c$; see the picture below. So, consider the points of the form $tC+(1-t)P$. Then\begin{align}\bigl|tC+(1-t)P-C\bigr|=9&\iff|(3+4i)(t-1)|=9\\&\iff|t-1|=\frac95\\&\iff t=\pm\frac{14}5\text{ or }t=-\frac45.\end{align}Can you take it from here?

