If ` P -= ((1)/(x_(p)), p ), Q = ((1)/(x_(q)),q), R = ((1)/(x_(r)), r)`
where ` x _(k) ne 0 `, denotes the ` k^(th) ` terms of a H.P. for ` k in N `, then :

Options

A. ar. `( Delta PQR)= (p^(2)q^(2)r^(2))/(2) sqrt((p-q)^(2) + (q-r)^(2) + (r-p)^(2))`
B. ` Delta PQR` is a right angled triangle
C. the points P,Q, R are collinear
D. None of these

Answer

To solve the problem, we need to analyze the coordinates of points P, Q, and R, which are given as:

- \( P = \left( \frac{1}{x_p}, p \right) \)
- \( Q = \left( \frac{1}{x_q}, q \right) \)
- \( R = \left( \frac{1}{x_r}, r \right) \)

where \( x_k \neq 0 \) denotes the \( k^{th} \) term of a harmonic progression (H.P.) for \( k \in \mathbb{N} \).

### Step 1: Understanding Harmonic Progression
If \( \frac{1}{x_1}, \frac{1}{x_2}, \ldots, \frac{1}{x_k} \) are in H.P., then their reciprocals \( x_1, x_2, \ldots, x_k \) are in Arithmetic Progression (A.P.).

### Step 2: Expressing the Terms in A.P.
Let’s denote the first term of the A.P. as \( a_1 \) and the common difference as \( d \). Then we can express:
- \( x_p = a_1 + (p - 1)d \)
- \( x_q = a_1 + (q - 1)d \)
- \( x_r = a_1 + (r - 1)d \)

### Step 3: Finding the Coordinates
Now substituting these into the coordinates of points P, Q, and R:
- \( P = \left( \frac{1}{a_1 + (p - 1)d}, p \right) \)
- \( Q = \left( \frac{1}{a_1 + (q - 1)d}, q \right) \)
- \( R = \left( \frac{1}{a_1 + (r - 1)d}, r \right) \)

### Step 4: Finding the Slopes
Next, we need to find the slopes of the lines connecting these points:
1. **Slope of PQ**:
\[ \text{slope of } PQ = \frac{q - p}{\frac{1}{a_1 + (q - 1)d} - \frac{1}{a_1 + (p - 1)d}} = \frac{q - p}{\frac{(a_1 + (p - 1)d) - (a_1 + (q - 1)d)}{(a_1 + (p - 1)d)(a_1 + (q - 1)d)}} \]
Simplifying gives:
\[ = \frac{(q - p)(a_1 + (p - 1)d)(a_1 + (q - 1)d)}{(q - p)d} \]

2. **Slope of QR**:
\[ \text{slope of } QR = \frac{r - q}{\frac{1}{a_1 + (r - 1)d} - \frac{1}{a_1 + (q - 1)d}} \]

3. **Slope of PR**:
\[ \text{slope of } PR = \frac{r - p}{\frac{1}{a_1 + (r - 1)d} - \frac{1}{a_1 + (p - 1)d}} \]

### Step 5: Conclusion
Since all slopes are equal, we conclude that points P, Q, and R are collinear. Therefore, the correct answer is that the area of triangle PQR is zero, which corresponds to option 3.