Straight Lines
Straight Lines
nta_pyq_2025_jan
Grade 11
Question:
Let n Cr-1 = 28, n Cr = 56 and n Cr+1 = 70 . Let A(4 cos t, 4 sin t), B(2 sin t, -2 cos t) and C (3r - n, r - n - 1) 2 be the vertices of a triangle ABC , where t is a parameter. If (3x - 1) + (3y) = \alpha, is the locus of the centroid of 2 2 triangle ABC , then \alpha equals
Step-by-Step Solution
Key Concept: Apply the core result for lines, slopes and distances and simplify using the given constraints.
Cr-1 = 28 ⎫ ⎪ ⎪ ⎪ n ⎪ ⎪ (4) Cr = 56 ⎪ ⎪ ⎪ ⎪ n ⎪ Cr+1 = 70 ⎪ n ⎬n = 8 = 3 Cr-1 28 r 1 = \Rightarrow = ⎪ ⎪ n 56 n-r+1 2 ⎪ ⎪ Cr ⎪ ⎪ ⎪ n ⎪ ⎪ Cr 56 r+1 70 ⎪ ⎪ ⎭ n = \Rightarrow = Cr+1 70 n-r 56 4 cos t+2 sin t+1 4 sin t-2 cos t h = k = 3 3 3h - 1 = 4 cos t + 2 sin t 3k - 1 = 4 sin t - 2 cos t 2 2 (1) + (2) 2 2 (3h - 1) + (3k) = 20 Locus of centroid: (3x - 1) + (3y) = 20 2 2 \Rightarrow \alpha = 20
Correct Answer: 4