Vector Algebra
Vector Algebra
nta_pyq_2025_jan
Grade 12
Question:
Let A, B, C be three points in xy-plane, whose position vector are given by \sqrt3^i + ^j, ^i + \sqrt3^j and a^i + (1 - a)^j respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between - -\to - -\to the vectors OA and OB is , then the sum of all the possible values of a is : 9 \sqrt2
Step-by-Step Solution
Key Concept: Apply the core result for dot product, cross product and projections and simplify using the given constraints.
Equation of line in the internal bisector of OA˙ and OB is (\sqrt3 + 1)^i + (\sqrt3 + 1)^j (3) \Rightarrow line will be y = x \Rightarrow x - y = 0 ∣ a-(1-a) ∣ 9 D = ∣ ∣ = ∣ \sqrta2 +(1-a)2 ∣ \sqrt2 81 2 2 2 (2a - 1) = (a + (1 - a) ) 2 2 2 2 \Rightarrow 2 (4a - 4a + 1) = 81a + 81a - 162a - 81 2 2 \Rightarrow 162a - 162a + 81 - 8a + 8a - 2 = 0 2 \Rightarrow 154a - 154a + 79 = 0 (-154) Sum of values = - = 1 154
Correct Answer: 3