Straight Lines
Straight Lines
nta_pyq_2025_jan
Grade 11
Question:
Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AM N is 4 9 of the area of the triangle OAB and AN : NB = \lambda : 1, then the sum of all possible value(s) of is \lambda :
Step-by-Step Solution
Key Concept: Apply the core result for lines, slopes and distances and simplify using the given constraints.
(1) 1 Area of △AOB = 2 4 1 2 Area of △AMN = \times = 9 2 9 Equation of AB is x + y = 1 ∘ OA = 1, AM = sec(45 - \theta) ∘ AN = sec(45 - \theta) cos \theta ∘ MN = sec(45 - \theta) sin \theta 1 2 2 ∘ Ar(△AMN) = \times sec (45 - \theta) sin \theta ⋅ cos \theta = 2 9 1 \Rightarrow tan \theta = 2, 2 tan \theta = 2 is rejected AN \lambda = = cot \theta = 2 NB 1
Correct Answer: 1