Straight Lines
Statement-Based — Orthocentre and Concurrent Lines
nta_pyq_2026_jan
Grade None
Question:
Among the statements:
(S1): If $A(5,-1)$ and $B(-2,3)$ are two vertices of a triangle, whose orthocentre is $(0,0)$, then its third vertex is $(-4,-7)$.
(S2): If positive numbers $2a,b,c$ are three consecutive terms of an A.P., then the lines $ax+by+c=0$ are concurrent at $(2,-2)$.
both are incorrect
only (S2) is correct
only (S1) is correct
both are correct
Step-by-Step Solution
Key Concept: S1: $m_{AO}\cdot m_{BC}=-1$ and $m_{AB}\cdot m_{OC}=-1$. From $m_{AO}\cdot m_{BC}=-1$: $5h-k+13=0$. From $m_{AB}\cdot m_{OC}=-1$: $4k=7h$. Solving: $h=-4$, $k=-7$. S1 correct. S2: AP means $b=2a+c/2$... $2a-2b+c=0\Rightarrow$ passes through $(2,-2)$. S2 correct.
Both S1 and S2 are correct.
Correct Answer: 4