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.
$\therefore \frac{2a+0+c}{\sqrt{a^2+b^2}}+\frac{0+2b+c}{\sqrt{a^2+b^2}}+\frac{a+b+c}{\sqrt{a^2+b^2}}=0$
$\Rightarrow 3a+3b+3c=0$
$\Rightarrow a+b+c=\mathrm{0...}(ii)$
From Eqs. (i) and (ii) $ax+by+c=0$ always passes
through a fixed point $(1,1)$.