<p>If 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 ≠ 0, denotes the kth terms of a H.P. for k ∈ ℕ, then:</p>
<p>(a) ar(△PQR) = \(\frac{p^2 q^2 r^2}{2}[(p-q)^2 + (q-r)^2 + (r-p)^2]\)</p>
<p>(b) △PQR is a right angled triangle</p>
<p>(c) The points P, Q, R are collinear</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: If x_k denotes the kth term of a H.P., then 1/x_k forms an A.P. This means the y-coordinates p, q, r (which equal 1/x_p, 1/x_q, 1/x_r respectively in the A.P.) satisfy a linear relationship with the x-coordinates.
<p><strong>Step 1: Understanding H.P. and A.P. relationship</strong></p><p>If x_k denotes the kth term of a H.P., then the sequence {1/x_k} forms an A.P. with some first term a and common difference d.</p><p>Therefore: 1/x_k = a + (k-1)d</p><p><strong>Step 2: Express coordinates of P, Q, R</strong></p><p>Given: P ≡ (1/x_p, p), Q ≡ (1/x_q, q), R ≡ (1/x_r, r)</p><p>Since 1/x_k is the kth A.P. term: </p><p>• 1/x_p = a + (p-1)d, so first coordinate of P is a + (p-1)d</p><p>• 1/x_q = a + (q-1)d, so first coordinate of Q is a + (q-1)d</p><p>• 1/x_r = a + (r-1)d, so first coordinate of R is a + (r-1)d</p><p>The second coordinates are p, q, r respectively.</p><p><strong>Step 3: Check collinearity using slope condition</strong></p><p>Let x-coordinate be u = 1/x_k and y-coordinate be v = k.</p><p>For point P: (u_p, v_p) = (a + (p-1)d, p)</p><p>For point Q: (u_q, v_q) = (a + (q-1)d, q)</p><p>For point R: (u_r, v_r) = (a + (r-1)d, r)</p><p><strong>Step 4: Calculate slope between P and Q</strong></p><p>Slope PQ = (q - p)/(a + (q-1)d - a - (p-1)d) = (q - p)/((q-p)d) = 1/d</p><p><strong>Step 5: Calculate slope between Q and R</strong></p><p>Slope QR = (r - q)/(a + (r-1)d - a - (q-1)d) = (r - q)/((r-q)d) = 1/d</p><p><strong>Step 6: Verify collinearity</strong></p><p>Since slope PQ = slope QR = 1/d, all three points lie on the same straight line.</p><p>The equation of the line is: v - p = (1/d)(u - (a + (p-1)d))</p><p>Simplifying: dv - dp = u - a - (p-1)d</p><p>Therefore: u = a + dv - d</p><p>Or: 1/x_k = a + dk - d, which confirms all three points satisfy this linear relation.</p><p><strong>∴ Answer:</strong> c</p>
Correct Answer: c