Analysis:

To solve this problem, we need to understand the motion of the ball and the forces acting on it.

1. Forces Acting on the Ball:

The ball is suspended by a thread, so there are two main forces acting on it:
- Tension force (T): The thread exerts a tension force on the ball, always directed towards the center of the circular motion.
- Weight force (W): The force of gravity acting on the ball, always directed vertically downward.

2. Acceleration at Extreme and Lowest Positions:

At the extreme positions (highest and lowest points), the acceleration of the ball is equal. Let's denote this acceleration as 'a'.

At the highest point, the acceleration is directed towards the center of the circular motion. At the lowest point, the acceleration is directed away from the center of the circular motion.

3. Centripetal Force:

The tension force acting on the ball provides the necessary centripetal force for circular motion. The centripetal force is given by the equation:

T = (m * v^2) / r

where m is the mass of the ball, v is the velocity of the ball, and r is the radius of the circular motion.

4. Equating Forces:

At the highest point, the weight force and tension force add up to provide the net force required for circular motion. At the lowest point, the weight force and tension force subtract to provide the net force required for circular motion.

5. Applying Equations:

Considering the forces acting on the ball at the highest point:
- Net force = T - W
- Centripetal force = m * a

Setting these two equations equal to each other, we get:

T - W = m * a

Considering the forces acting on the ball at the lowest point:
- Net force = T + W
- Centripetal force = m * a

Setting these two equations equal to each other, we get:

T + W = m * a

6. Solving the Equations:

We can solve these two equations simultaneously to find the acceleration 'a' and the tension force 'T'. Once we know the tension force, we can determine the deflection angle 'θ' using trigonometry.

7. Final Calculation:

Given that the acceleration values at the extreme positions are equal, we can solve the equations to find the deflection angle 'θ'.

The correct answer is '53 degrees'.