If (`a x^2+c)y+(a x^2+c )=0` and `x` is a rational function of `y` and `ac` is negative, then

Options

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

Answer

2

Explanation

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