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 $( - 4, - 7)$ 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 ( $2, - 2$ ),
Options
Solution
Solution of statement-1
$$m_{AO} \cdot m_{BC} = - 1 $$
$$\begin{array}{r} \Rightarrow 5\text{ }h - k + 13 = 0\#(1) \end{array}$$
& $m_{AB} \cdot m_{OC} = - 1$
$$\begin{array}{r} \Rightarrow 4k = 7\text{ }h\#(2) \end{array}$$
⇒ third vertex is $( - 4, - 7)$
∴ Statement 1 is correct.
Solution of statement-2
$2a,b,c \rightarrow$ A.P.
$${b = \frac{2a + c}{2} }{\Rightarrow 2a - 2\text{ }b + c = 0 }$$∵ lines $ax + by + c = 0$ 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.
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