C

Step 1: To find the speed of the particle at the distance from the center of the ring, we can apply the concepts of gravitational potential energy and kinetic energy. The gravitational potential energy (U) at a distance x from the center is given by:
U = -\frac{G M m}{\sqrt{R^2 + x^2}}
where M is the mass of the ring, m is the mass of the point particle, G is the universal gravitational constant, R is the radius of the ring, and x is the vertical distance from the center of the ring.

Step 2: The kinetic energy (K) of the particle when it reaches the lowest point (x = 0) is given by: K = \frac{1}{2} m v^2
where v is the velocity of the particle. The conservation of mechanical energy states that the total mechanical energy remains constant, so we can equate the gravitational potential energy at the starting point to the kinetic energy at the lowest point.
So, -\frac{G M m}{\sqrt{R^2 + x^2}} = \frac{1}{2} m v^2

Step 3: By rearranging and solving for v, we get:
v = \sqrt{\frac{2 G M}{\sqrt{R^2 + x^2}}}
To express this equation in terms of r (distance from the center of the ring), substitute x = h:
v = \sqrt{\frac{2 G M}{h}}

Therefore, C is the correct answer.