Question

Let the algebraic sum of the perpendicular distances from the points $(2,0),(0,2)$ & $(1,1)$ to a variable straight line be zero, then the line passes through a fixed point whose co-ordinates are

• A. $(1,1)$
• B. $(2,3)$
• C. $\left(\frac{3}{5},\frac{3}{5}\right)$
• D. None of these

Answer 1

The correct option is A $(1,1)$

Let the variable straight line be $ax+by+c=0$ ...(1)
where algebraic sum of perpendiculars from $(2,0),(0,2)$ and $(1,1)$ is zero

$\therefore \frac{2a+0+c}{\sqrt{a^2+b^2}}+\frac{0+2b+c}{\sqrt{a^2+b^2}}+\frac{a+b+c}{\sqrt{a^2+b^2}}=0$

$\Rightarrow 3a+3b+3c=0\Rightarrow a+b+c=0$
From equation (1) and (2) $ax+by+c=0$ always passes through a fixed point $(1,1).$