Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 63 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.


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Problem

If any one pair among the straight lines
ax + by = a + b, bx –(a + b)y = – a, (a + b)x –ay = b
intersect, then show that the three straight lines are concurrent.


Solution:

Three lines are concurrent if each of them is linear combination of other two & they are not parallel.
Now given one pair intersect that is they are not parallel.
Now
ax + by = a + b … (i)
bx – (a + b)y = – a …(ii)
(a +b)x –ay = b …(iii)
(i) + (ii) – (iii) = 0
So they are concurrent.