Straight Lines
Straight Lines
nta_pyq_2025_jan
Grade 11
Question:
A rod of length eight units moves such that its ends A and B always lie on the lines x - y + 2 = 0 and y + 2 = 0 , respectively. If the locus of the point P , that divides the rod AB internally in the ratio 2 : 1 is 9 (x 2 + \alphay 2 + \betaxy + \gammax + 28y) - 76 = 0 , then \alpha - \beta - \gamma is equal to :
Step-by-Step Solution
Key Concept: Apply the core result for lines, slopes and distances and simplify using the given constraints.
AB = 8 2 (3) AB = 64 \Rightarrow (a - b) 2 + (b + 4) 2 = 64 ...(1) Now P divides AB in the ratio 2 : 1 internally \Rightarrow h = 2a+b 3 and k = -4+b+2 3 \Rightarrow 2a + b = 3h \ldots (2) k = b-2 3 From equation (2) and (3) \Rightarrow b = 3k + 2 \Rightarrow 2a = 3h - 3k - 2 3h - 3k - 2 \Rightarrow a = 2 Now by putting value of a and b in equation 2 3h - 3k - 2 2 \Rightarrow ( - (3k + 2)) + (3k + 2 + 4) = 64 2 2 3h - 3k - 2 - 6k - 4 2 \Rightarrow ( ) + (3k + 6) = 64 2 2 2 \Rightarrow (3h - 9k - 6) + 4(3k + 6) = 4 \times 64 2 2 \Rightarrow 9(h - 3k - 2) + 36(k + 2) = 256 2 2 \Rightarrow 9 (h + 9k + 4 - 6hk - 4h + 12k) 2 + 36 (k + 4 + 4k) = 256 2 2 \Rightarrow 9 (h + 13k + 20 - 6hk - 4h + 28k) = 256 Replacing h by x and k by y 2 2 \Rightarrow 9 (x + 13y - 6xy - 4x + 28y) + 180 - 256 = 0 2 2 \Rightarrow 9 (x + 13y - 6xy - 4x + 28y) - 76 = 0 By comparing \alpha = 13, \beta = -6, \gamma = -4 \alpha - \beta - \gamma = 13 + 6 + 4 = 23
Correct Answer: 3