To solve this problem, we can use the concept of proportionality and the given ratio of the broken down diamond. Let's break down the problem into steps for better understanding:

Step 1: Understanding the given information
- The price of a diamond is directly proportional to the square of its weight. This means that if we double the weight of a diamond, its price will become four times.
- The diamond was broken down in the ratio 5:2:7. This means that the original weight of the diamond can be represented as 5x + 2x + 7x, where x is a common factor.

Step 2: Finding the ratio of weights
- The total weight of the diamond after being broken down is 5x + 2x + 7x = 14x.
- Now, we need to find the individual weights in terms of x. Dividing the total weight by the ratio, we get:
- Weight 1 = (5x/14x) * 14x = 5x
- Weight 2 = (2x/14x) * 14x = 2x
- Weight 3 = (7x/14x) * 14x = 7x

Step 3: Understanding the loss
- The person faces a loss of Rs 11800. This means that the total price of the broken down diamond is Rs 11800 less than the initial price.
- Let's assume the initial price of the diamond is P. So, the total price of the broken down diamond is P - 11800.

Step 4: Applying the proportionality
- According to the given information, the price of the diamond is directly proportional to the square of its weight. Let's represent this proportionality mathematically:
- Initial Price / Total Weight^2 = Constant
- P / (14x)^2 = Constant
- P = Constant * (14x)^2

Step 5: Solving the equation
- We can now substitute the individual weights and the total price of the broken down diamond into the equation:
- P - 11800 = Constant * [(5x)^2 + (2x)^2 + (7x)^2]
- P - 11800 = Constant * (25x^2 + 4x^2 + 49x^2)
- P - 11800 = Constant * 78x^2

Step 6: Finding the constant
- Since the constant is the same in both equations, we can equate them:
- Constant * (14x)^2 = Constant * 78x^2
- (14x)^2 = 78x^2
- 196x^2 = 78x^2
- 196x^2 - 78x^2 = 0
- 118x^2 = 0

Step 7: Finding the value of x
- Dividing both sides of the equation by x^2, we get:
- 118 = 0
- This equation is not possible, which means that x must be 0.
- However, x cannot be 0 as it represents the common factor of the weights.
- Hence, there is an error in the calculation, and we need to recheck our steps.

After rechecking the steps, we find that there was an error in Step 6. The