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

Options

• (1, 1)

• (2, 1)

• (1, 2)

• none of these

Answer 1

(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.

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

\[ \Rightarrow 3a + 3b + 3c = 0\]

\[ \Rightarrow a + b + c = 0\]

Substituting c = \[-\]a \[-\]b in ax + by + c = 0, we get:

\[ax + by - a - b = 0\]

\[ \Rightarrow a\left( x - 1 \right) + b\left( y - 1 \right) = 0\]

\[ \Rightarrow \left( x - 1 \right) + \frac{b}{a}\left( y - 1 \right) = 0\]

This line is of the form

\[L_1 + \lambda L_2 = 0\],  which passes through the intersection of  \[L_1 = 0\text { and } L_2 = 0,\]  i.e.

x \[-\] 1 = 0 and y \[-\] 1 = 0.

\[\Rightarrow\] x = 1, y = 1