<p><strong>511.</strong> \(D\)-\(ABC\) is a tetrahedron with \(A = (2, 0, 0)\), \(B = (0, 4, 0)\) and \(CD = \sqrt{14}\). Edge \(CD\) lies on the line \(\dfrac{x-1}{1} = \dfrac{y-1}{2} = \dfrac{z-3}{3}\). If locus of centroid of tetrahedron is \(\dfrac{x - \dfrac{3}{2}}{1} = \dfrac{y - y_1}{a} = \dfrac{z - z_1}{b}\), then which of the following is/are true:</p>
Step-by-Step Solution
Key Concept: The centroid of a tetrahedron is the average of its four vertices: G = (A+B+C+D)/4. Since C and D lie on a fixed line but vary, the centroid traces a line parallel to that line. The direction ratios of the locus equal the direction ratios of line CD.
Step 1: Centroid of tetrahedron D-ABC is G = (A+B+C+D)/4 where A=(2,0,0), B=(0,4,0). Step 2: Points C and D lie on line: (x-1)/1 = (y-1)/2 = (z-3)/3. Parametrize: C = (1+s, 1+2s, 3+3s) and D = (1+t, 1+2t, 3+3t) for parameters s, t. Step 3: Centroid: G = [(2+0+1+s+1+t)/4, (0+4+1+2s+1+2t)/4, (0+0+3+3s+3+3t)/4] G = [(4+s+t)/4, (6+2s+2t)/4, (6+3s+3t)/4] G = [1 + (s+t)/4, 3/2 + (s+2t)/4, 3/2 + 3(s+t)/4] Step 4: Let u = s+t (the parameter for locus). As s, t vary independently over ℝ, u also varies over ℝ: G = (1 + u/4, 3/2 + u/2, 3/2 + 3u/4) Step 5: Direction ratios of locus are proportional to (1/4, 1/2, 3/4) = (1, 2, 3)—matching CD's direction ratios (1, 2, 3). Step 6: When u=0: G passes through (1, 3/2, 3/2). So y_1 = 3/2, z_1 = 3/2, a = 2, b = 3. ∴ The locus is (x-1)/1 = (y-3/2)/2 = (z-3/2)/3, confirming answer A
Correct Answer: A