Question

If the algebraic sum of the perpendicular distances from the points $(2,0),(0,2)$ and $(1,1)$ to a variable straight line be zero, then the line passes through the point

• A. $(-1,1)$
• B. $(1,1)$
• C. $(1,-1)$
• D. $(-1,-1)$

Answer 1

Answer: (B)

$(1,1)$

Let the variable line be $ax+by+c=0$

Given, the algebraic sum of the perpendicular from the points (2,0),(0,2) and (1,1) to this line is zero

$\therefore \left|\frac{2\times a+b\times 0+c}{\sqrt{a^2+b^2}}\right|+\left|\frac{a\times 0+b\times 2+c}{\sqrt{a^2+b^2}}\right|+\left|\frac{a\times 1+b\times 1+c}{\sqrt{a^2+b^2}}\right|=0$

$\Rightarrow \pm \left(\frac{2a+c}{\sqrt{a^2+b^2}}\right)\pm \left(\frac{2b+c}{\sqrt{a^2+b^2}}\right)\pm \left(\frac{a+b+c}{\sqrt{a^2+b^2}}\right)=0$

$\Rightarrow 2a+c+2b+c+a+b+c=0$

$\Rightarrow 3a+3b+3c=0$

$\Rightarrow a+b+c=0$

This is a linear relation between a,b and c.

So, the equation $ax+by+c=0$ represents a family of straight line passing through a fixed point.

Comparing $ax+by+c=0$ and $a+b+c=0$

We obtain

The coordinates of fixed point are $(1,1)$.