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

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). [2026]

Smart Achievers · Accepted Answer

(1)

$Solution of statement-1$

$m_{A O} \cdot m_{B C} = - 1$

$B (- 2, 3)$

$\Rightarrow 5 h - k + 13 = 0 . . . (1)$

$& m_{A B} \cdot m_{O C} = - 1$

$\Rightarrow 4 k = 7 h . . . (2)$

$\Rightarrow third vertex is (- 4, - 7)$

$∴$ $Statement 1 is correct.$

$Solution of statement-2$

$2 a, b, c \to A.P.$

$b = \frac{2 a + c}{2}$

$\Rightarrow 2 a - 2 b + c = 0$

$∵$ $lines a x + b y + c = 0 are concurrent then$

$\frac{x}{2} = \frac{y}{- 2} = \frac{1}{1}$

$x = 2 and y = - 2$

$∴$ $Point of concurrency is (2, - 2)$

$∴$ $Statement 2 is correct.$

Want Us to Email you About Special Offers & Updates?