Question

Let the algebraic sum of the perpendicular distance from the points $(2, 0) , (0, 2)$ and $(1, 1)$ to a variable straight line be zero, then the line passes through a fixed point whose coordinates are ....

Answer 1

Let the variable straight line be $ax + by + c = 0 $ ...(i)
where, algebraic sum of perpendiculars from $(2, 0), (0,2)$ and $(1, 1)$ is zero.
$\therefore \frac{2a + 0 + c}{\sqrt{a^2+b^2}}+\frac{0 + 2b + c}{\sqrt{a^2+b^2}}+\frac{a + b + c}{\sqrt{a^2+b^2}}=0$
$\Rightarrow 3a + 3b + 3c = 0$
$\Rightarrow a + b + c = 0 ...(ii)$
From Eqs. (i) and (ii) $ax + by + c = 0$ always passes
through a fixed point $(1, 1)$.