Straight Lines
Straight Lines
nta_pyq_2025_jan
Grade 11
Question:
Let A(6, 8), B(10 cos \alpha, -10 sin \alpha) and C(-10 sin \alpha, 10 cos \alpha), be the vertices of a triangle. If L(a, 9) and G(h, k) be its orthocenter and centroid respectively, then (5a - 3h + 6k + 100 sin 2\alpha) is equal to ______ -.
Step-by-Step Solution
Key Concept: Apply the core result for lines, slopes and distances and simplify using the given constraints.
(145) a + 0 = h \Rightarrow a = 3h 3 9 + 0 = k \Rightarrow k = 3 3 6 + 10 cos \alpha - 10 sin \alpha 8 - 10 sin \alpha - 10 cos \alpha ∵ (h, k) = ( , ) 3 3 6 + 10 cos \alpha - 10 sin \alpha = 3h 10 cos \alpha - 10 sin \alpha = 3h - 6 10(cos \alpha - sin \alpha) = 1 8 - 10 sin \alpha + 10 cos \alpha = k 3 \Rightarrow 100 sin 2\alpha = 99 7 h = 3 \Rightarrow a = 7 Now, 5a - 3h + 6k + 100 sin 2\alpha = 35 - 7 + 18 + 99 = 145
Correct Answer: 145