Question

Mark the correct alternative in each of the following :

L is a variable line such that the algebraic sum of the distance of the points $(1,1),(2,0)$ and $(0,2)$ from the line is equal to zero. The line L will always pass through

• A. (1, 1)
• B. (2, 1)
• C. (1, 2)
• D. none of these

Answer 1

The correct option is A

(1, 1)

Let $ax+by+c=0$ be tje variable line. It is that the algebraic sum of the distances of the points $(1,1),(2,0)$ and $(0,2)$ from the line is equal to zero.

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

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

$\Rightarrow a+b+c=0$

Substituting $c=-a-b$ in $ax+by+c=0,$ we get :

$ax+by-a-b=0$

$\Rightarrow a(x-1)+b(y-1)=0$

$\Rightarrow (x-1)+\frac{b}{a}(y-1)=0$

This line is of the form $L_1+\lambda L_2=0,$ which passes through the intersection of $L_1=0$ and $L_2=0,$ i.e. $x-1=0$ and $y-1=0$ \

$\Rightarrow x=1,y=1$