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.
∴ \(\frac{a+b+c}{\sqrt{a^2+b^2}}\)+ \(\frac{2a+0+c}{\sqrt{a^2+b^2}}\) + \(\frac{0+2b+c}{\sqrt{a^2+b^2}}\) = 0
⇒ 3a + 3b + 3c = 0
⇒ a + b + c = 0
Substituting c = – a – b in ax + by + c = 0, we get:
ax + by – a – b = 0
⇒ a(x – 1) + b(y – 1) = 0
⇒ x - 1 + \(\frac{b}{a}\)(y - 1) = 0
This line is of the form L1 + λL2 = 0, which passes through the intersection of L1 = 0 and L2 = 0, i.e. x – 1 = 0 and y – 1 = 0.
⇒ x = 1, y = 1
