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) and (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),
(A)Only (S1) is correct
(B)Only (S2) is correct
(C)Both are incorrect
(D)Both are correct✓ Correct
Answer: (D)
Solution
For Statement (S1), let the third vertex be C(h,k) with orthocentre at the origin. The altitude conditions OA⊥BC and OB⊥AC give −5h+k−13=0 and 2h−3k−13=0; solving, h=−4,k=−7, so the third vertex is (−4,−7). For Statement (S2), 2a,b,c in A.P. means 2b=2a+c, i.e. 2a−2b+c=0; substituting c=2b−2a into ax+by+c=0 gives a(x−2)+b(y+2)=0, which is satisfied by (2,−2) for all such a,b, so the lines are concurrent at (2,−2). Both statements are correct.
JEE Main · 2026 · 22 Jan 2026 · Shift 2 · Single correct option
