Straight Lines
Straight Lines
nta_pyq_2025_jan
Grade 11
Question:
Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2y = 2. If the centroid of △PQR is the point (\alpha, \beta), then 15(\alpha - \beta) is equal to :
Step-by-Step Solution
Key Concept: Apply the core result for lines, slopes and distances and simplify using the given constraints.
The centroid G (\alpha, \beta) of △P QR be image of centroid of given triangle P Q R . ′′ ′ ′ ′ (4) Centroid of $\Delta$P Q R = ( ′ ′ ′ 1+3+2 3 , 3+1+4 3 ′ ) = G (2, 8 3 ) . Image of G (2, 8 3 ) , w.r.t. line x + 2y = 2 is (\alpha, \beta) 8 16 \beta- -2(2+ -2) Then \alpha-2 3 3 = = 1 2 1+4 8 \beta- \alpha-2 3 32 \therefore = = - 1 2 15 and \beta = - 2 8 \therefore \alpha = - 15 5 Then 15(\alpha - \beta) = 15 (- 2 15 + 24 15 ) = 22
Correct Answer: 4