Which option is correct?

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2 Answers
May 14, 2017

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.

May 14, 2017

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.