The number of ordered pairs (a,b) such that the equations ax+by=1 and x2 +y2 =50 have all solutions integral ???
11(x,y) ≡ (±5,±5)
(±1,±7)
(±7,±1)
(a,b) ≡ (0,±1/5)
(±1/5,0)
(±1/10,±1/10)
(2/5,-1/5)
(-1/5,2/5)
(3/5,-2/5)
(-2/5,3/5)
(4/5,-3/5)
(-3/5,4/5)
(6/5,-1)
(-1,6/5) etc.
are only a drop in the ocean [4][4]
Ans : ∞ (infinite)
11@NISHANT SIR
what is the proper way to solve these type of probs
i tried it but not useful..............
62I think that it should have been given a, b are also integers..
and for that question answer wud have been zero
but since mona din ask any of those I resisted from replying
62@manipal...
the first thing to be done is that the RHS should be such that x and y are both perfect squares..
the only 2 ways are
(1,7) and (5,5)
with all + - and interchanges..
then the second conclusion wud be trivial..
take any value of a, for that value you will get a vlaue of b.
hence infinte solutoins.
1just to add up...
we can see it in this way like:
(|x|, |y|) can be (5,5) and (1,7) or (7,1) ...
so from ax+by=1
we get lines 5a+5b=1 and 7a+b=1 etc etc... all those... where the variables are a and b.
so infinite no of real values of a and b satisfies.
11but the options given in the ques. are
76
45
66
89
which one is correct??
62mona.. is something given about a,b being rational or integers..??
see the question carefully...