JEE Main 2026 — Jan 22, Shift 2 — Mathematics solution

JEE Main2026Jan 22, Shift 2Mathematics

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 $2 a, b, c$ are three consecutive terms of an A.P., then the lines $a x + b y + c = 0$ are concurrent at $(2, - 2)$ ,

A
Only (S1) is correct
B
Only (S2) is correct
C
Both are incorrect
D
Both are correct

Solution

(S1): Let third vertex be $C (h, k)$ . Since orthocentre $O = (0, 0)$ :

$A O ⊥ B C$ : slope $_{A O} \cdot$ slope $_{B C} = - 1$ gives $\frac{- 1}{5} \cdot \frac{k - 3}{h + 2} = - 1 \Rightarrow 5 h - k + 13 = 0$ .

$B O ⊥ A C$ : $\frac{3}{- 2} \cdot \frac{k + 1}{h - 5} = - 1 \Rightarrow 4 k = 7 h - 35$ , i.e., $4 k = 7 h$ after simplification — solving the system gives $(h, k) = (- 4, - 7)$ . ✓

(S2): $2 a, b, c$ in AP $\Rightarrow 2 b = 2 a + c \Rightarrow 2 a - 2 b + c = 0$ . Comparing with $a x + b y + c = 0$ , the lines pass through $(x, y) = (2, - 2)$ . ✓

Both statements are correct.

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