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.

Hint
I have to admit that I don't really understand what you did...
However, as you noticed the question is about finding the minimum and maximum distance between the point $p=2+3i$ and a point $z$ lying on the circle $\mathcal C$ of center $c=5+7i$ with a radius equal to $9$.
And the minimum and maximum distances are attained on the 2 points of intersection between the circle $\mathcal C$ and the line $\mathcal L$ passing through $p$ and $c$. A point of $\mathcal L$ is a complex $Z= (5+7i) + t [(5+7i)-(2+3i)]$ where $t \in \mathbb R$.
Plugging in $Z$ in the equation of the circle, you should find two values for  $t$ corresponding on the line to the extremas for the distance.
