Question

L is a variable line such that the algebraic sum of the distances 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

(a) (1,1)

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

$$\begin{gathered} \therefore \frac{a+b+c}{\sqrt{a^2+b^2}}+\frac{2a+0+c}{\sqrt{a^2+b^2}}+\frac{0+2b+c}{\sqrt{a^2+b^2}}=0 \\ \Rightarrow 3a+3b+3c=0 \\ \Rightarrow a+b+c=0 \end{gathered}$$

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

$$\begin{gathered} ax+by-a-b=0 \\ \Rightarrow a\left(x-1\right)+b\left(y-1\right)=0 \\ \Rightarrow \left(x-1\right)+\frac{b}{a}\left(y-1\right)=0 \end{gathered}$$

This line is of the form $L_1+\lambda L_2=0$, which passes through the intersection of $L_1=0\mathrm{and}{L}_2=0,$ i.e. x $-$ 1 = 0 and y $-$ 1 = 0.

$\Rightarrow$ x = 1, y = 1