Question

Which option is correct?

Answer 1

All of them.

Explanation:

By inspection, all of the terms contain an `x` or `y` thus `(0,0)` is a solution for all of them for any a or b. Even though option 4 is only a point `(0,0)` It counts as a rational solution.

Answer 2

Option `1` and `3` only.

Explanation:

Expand:
`a^2y^2-2abxy+b^2x^2+4xy=0`
`(b^2)x^2+(4y-2aby)x+(a^2y^2)=0`

Using discriminant:
`(4y-2aby)^2-4(b^2)(a^2y^2)=n^2`

This simplifies to:
`16y^2(1-ab)=n^2`

For the equation to get rational solutions, then the equation should be equal a perfect square. `16y^2` is already one then `(1-ab)` should be too.

Option `1` and `3` only since they equal to `0` and `1/4` respectively.