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If {a(x^2) + c}y + {a’(x^2) + c’} = 0 and x is a rational function of y and ac is negative,then 1)ac’ + a’c = 0 2)a/a’ = c/c’ 3)a^2 + c^2 = a’^2 + c’^2 4)aa’ + cc’ =1 Please explain the question..What does the question mean by x is a rational function of y??

If {a(x^2) + c}y + {a’(x^2) + c’} = 0 and x is a rational function of y and ac is negative,then 
1)ac’ + a’c = 0
2)a/a’ = c/c’
3)a^2 + c^2 = a’^2 + c’^2
4)aa’ + cc’ =1
Please explain the question..What does the question mean by x is a rational function of y??

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1 Answers

Aditya Gupta
2081 Points
5 years ago
rational function means a ratio of two polynomials. eg. (3x^3+x-1)/(x^3-7x) is a rational function.
now, we can write the eqn as
(ay+a’)x=root( –acy – y(a’c+ac’) – a’c’).......(1),
but x=p(y)/f(y), p and f being polynomials.
so we conclude that LHS of (1) is a rational function as well. which implies that RHS is a rational function too, which is possible iff –acy – y(a’c+ac’) – a’c’ is a perfect square, or discriminant is zero.
so, (a’c+ac’)^2=4aca’c’
or (a’c-ac’)^2=0
or a’c=ac’
or a/a’=c/c’

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