Straight Lines
Straight Lines
nta_pyq_2025_jan
Grade 11
Question:
Let the lines 3x - 4y - \alpha = 0, 8x - 11y - 33 = 0, and 2x - 3y + \lambda = 0 be concurrent. If the image of the point in the line 2x - 3y + \lambda = 0 is ( , then |\alpha\lambda| is equal to 57 -40 (1, 2) , ) 13 13
Step-by-Step Solution
Key Concept: Apply the core result for lines, slopes and distances and simplify using the given constraints.
(3) ∵ PM = QM 57 -40 ⎛ + 1 + 2 ⎞ 13 13 So, M , ⎝ 2 2 ⎠ 35 -7 = ( , ) 13 13 ∵ M lies on the time 2x - 3y + \lambda = 0 35 -7 2( ) - 3( ) + \lambda = 0 13 13 70 21 \lambda = - + 13 13 -91 = = -7 13 ∣3 -4 -\alpha ∣ ∣ ∣ 8 -11 -33 = 0 ∣ ∣ ∣2 3 \lambda ∣ \Rightarrow 3(-11\lambda - 99) + 4(8\lambda + 66) - \alpha(-24 + 22) = 0 \Rightarrow 33\lambda - 297 + 32\lambda + 264 + 24\alpha - 22\alpha = 0 \Rightarrow -\lambda + 2\alpha - 33 = 0 \therefore \lambda = -7 - (-7) + 2\alpha - 33 = 0 2\alpha = 26 \alpha = 13 \therefore |\alpha\lambda| = |13 \times (-7)| = 91 n
Correct Answer: 3