If (`a x^2+c)y+(a x^2+c )=0a n dx` is a rational function of `ya n da c` is negative, then `a c^(prime)+c^(prime)c=0` b. `a//a '=c//c '` c. `a^2+c^2=a^('2)+c^('2)` d. `a a^(prime)+cc^(prime)=1`

Options

A. `ac' + a' c = 0`
B. `(a)/(a')=(c)/(c')`
C. `a^(2) + c^(2) = a'^(2) + c'^(2)`
D. `aa' + cc' = 1`

Answer

B

Explanation

We have, `(ax^(2)+c) y + (a'x^(2) + c') = 0 rArr x^(2) (ay + a') + (cy + c') = 0` ltbtgt If x is rational, then the discriminant of the above equation must be a perfect square.
`i.e." "0-4(ay+a')(cy+c')` must be a perfect square
`i.e." "-acy^(2) - (ac'+ a'c) y - a'c'` must be a perfect square
`rArr" "(ac'+a'c)^(2) - 4ac a' c' = 0" "[because "Disc" = 0]`
`rArr" "(ac' - a'c)^(2) = 0 rArr ac' = a'c rArr (a)/(a') = (c)/(c')`