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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

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Solution

(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.

a+b+ca2+b2+2a+0+ca2+b2+0+2b+ca2+b2=03a+3b+3c=0a+b+c=0

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

ax+by-a-b=0ax-1+by-1=0x-1+bay-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

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