Identify the correct statement-

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(A) For option $(A)$: Let the line be $ax + by + c = 0$. The sum of perpendicular distances is $\frac{4a+2b+c}{\sqrt{a^2+b^2}} + \frac{2a+4b+c}{\sqrt{

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If the algebraic sum of lengths of perpendiculars drawn from $(4,2)$,$(2,4)$,and the origin to a variable line is zero,then the line necessarily passes through $(2,2)$.

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(A) For option $(A)$: Let the line be $ax + by + c = 0$. The sum of perpendicular distances is $\frac{4a+2b+c}{\sqrt{a^2+b^2}} + \frac{2a+4b+c}{\sqrt{a^2+b^2}} + \frac{c}{\sqrt{a^2+b^2}} = 0$. This implies $6a + 6b + 3c = 0$,or $2a + 2b + c = 0$. This is the condition for the line to pass through $(2,2)$. Thus,$(A)$ is true.For option $(B)$: The condition $Ar(\Delta PAB) + Ar(\Delta PBC) + Ar(\Delta PAC) = Ar(\Delta ABC)$ is satisfied for any point $P$ inside $\Delta ABC$,not just the orthocenter. Thus,$(B)$ is false.For option $(C)$: The angle bisectors of a pair of lines are perpendicular to each other,they do not bisect the angle between themselves. Thus,$(C)$ is false.For option $(D)$: The harmonic conjugate of the midpoint of $A$ and $B$ is at infinity. Thus,$(D)$ is false.

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