in this diagram an object is moving in uniform circular motion in given trajectory which is a part of a circle. It's asked to find a point where the normal force is the least. I have shown my solution in picture. I don't know if my approach is right or not.
You can think about the normal force as the force that keeps the object from "flying into the path" - so it has to balance out all the other force components in the radial direction. In the radial direction, given that the object is an angle $\alpha$ above the horizontal (same as in your image), there are two components:
Since the centrifugal force acts "out of the circle" and the component of the gravitational force acts "into the circle", the normal force has to balance out their difference. So the magnitude of the normal force is given by $N = m\omega^2r - mg\textrm{sin}(\alpha)$. Given that the object is in uniform circular motion, nothing changes in the centrifugal force component, so it is a constant. That means that to maximise the magnitude of the normal force, one has to minimise $mg\textrm{sin}(\alpha)$. And since $m$ and $g$ are constant, that is equivalent to minimising $\textrm{sin}(\alpha)$, which gives you the position 1.
So your approach was correct, and it could be generalised to situations where there are other forces acting on the object - you would just add their components to the equation of the normal force and find the point where it attains a maximum.
