To find the ratio of the radius of curvature at the highest point of the path to the maximum height, we need to analyze the motion of the particle and use the principles of projectile motion.

Projectile motion is the motion of an object thrown into the air at an angle to the horizontal. It can be divided into two independent motions: horizontal motion with constant velocity and vertical motion with constant acceleration due to gravity.

Let's break down the problem into steps:

Step 1: Analyze the horizontal and vertical components of motion.
- The initial velocity of the particle can be resolved into its horizontal and vertical components as follows:
- Horizontal component: v₀cosα
- Vertical component: v₀sinα

Step 2: Find the time of flight and maximum height.
- The time of flight (T) can be determined using the formula: T = 2v₀sinα/g, where g is the acceleration due to gravity.
- The maximum height (H) can be calculated using the formula: H = (v₀sinα)²/2g.

Step 3: Determine the radius of curvature at the highest point.
- The radius of curvature (r) at any point of the projectile's trajectory can be given by the formula: r = (v₀²/g)sin²α.
- At the highest point, the velocity is purely horizontal, and the radius of curvature can be calculated using the formula: r = (v₀²/g)sinα.

Step 4: Calculate the ratio of r to H.
- The ratio of r to H can be found by dividing r by H: r/H = [(v₀²/g)sinα] / [(v₀sinα)²/2g].
- Simplifying this expression, we get: r/H = 2cot²α.

So, the correct option is (a) 2cot²α.

To summarize:
- The ratio of the radius of curvature at the highest point of the particle's path to the maximum height is 2cot²α.
- This can be derived by analyzing the horizontal and vertical components of motion, finding the time of flight and maximum height, and using the formula for the radius of curvature at the highest point.
- It is important to understand the principles of projectile motion and use the appropriate formulas to solve such problems.