There are two sets of parallel lines, their equations being `x cos alpha+y sin alpha=p` and `x sin alpha- y cos alpha=p` , `p=1,2,3,….n` and `alpha in (0,pi//2)`. If the number of rectangles formed by these two sets of lines is `225`, then the value of `n` is equals to (a) 4 (b) 5 (c) 6 (d) 7

Options

A. `4`
B. `5`
C. `6`
D. `7`

Explanation

To solve the problem, we need to determine the value of \( n \) such that the number of rectangles formed by two sets of parallel lines is 225.

### Step-by-step Solution:

1. **Understanding the Problem**:
We have two sets of parallel lines given by the equations:
\[ x \cos \alpha + y \sin \alpha = p \]
\[ x \sin \alpha - y \cos \alpha = p \]
where \( p \) takes values from 1 to \( n \).

2. **Forming Rectangles**:
To form a rectangle, we need to choose 2 lines from the first set and 2 lines from the second set. The number of ways to choose 2 lines from \( n \) lines is given by the combination formula \( \binom{n}{2} \).

3. **Calculating the Number of Rectangles**:
The total number of rectangles formed is:
\[ \text{Number of rectangles} = \binom{n}{2} \times \binom{n}{2} \]
This can be expressed as:
\[ \text{Number of rectangles} = \left( \frac{n(n-1)}{2} \right)^2 \]

4. **Setting Up the Equation**:
We know from the problem statement that the number of rectangles is 225:
\[ \left( \frac{n(n-1)}{2} \right)^2 = 225 \]

5. **Taking Square Roots**:
Taking the square root of both sides gives:
\[ \frac{n(n-1)}{2} = 15 \]

6. **Multiplying Both Sides by 2**:
Multiplying both sides by 2 results in:
\[ n(n-1) = 30 \]

7. **Rearranging the Equation**:
Rearranging gives us the quadratic equation:
\[ n^2 - n - 30 = 0 \]

8. **Factoring the Quadratic**:
We can factor this quadratic as:
\[ (n - 6)(n + 5) = 0 \]

9. **Finding the Roots**:
Setting each factor to zero gives:
\[ n - 6 = 0 \quad \Rightarrow \quad n = 6 \]
\[ n + 5 = 0 \quad \Rightarrow \quad n = -5 \]

10. **Selecting the Valid Solution**:
Since \( n \) represents the number of lines, it must be a positive integer. Therefore, we discard \( n = -5 \) and accept:
\[ n = 6 \]

### Conclusion:
The value of \( n \) is \( 6 \).