Question

If $xy>0$, which of the following has to be true?
I. $x+y=0$
II. $2y-2x<0$
III. $x^2+y^2>0$

• A. I only
• B. III only
• C. I and III
• D. II and III

Answer 1

The correct option is B III only

Given, $xy>0⟹x>0$ and $y>0$ or $x<0$ and $y<0$

Adding both $x>0$ and $y>0$, we get

$x+y>0$ or $x+y<0$

$I.$ Hence, $x+y=0$ is false

Subtracting $x$ from $y$ we get,

$y-x>0$ or $y-x=0$ or $y-x<0$

$II.$ Hence, $2y-2x<0$ is not always true.

Squaring both, we get $x^2>0$ and $y^2>0$

Adding both $x^2>0$ and $y^2>0$

we get $x^2+y^2>0$

$III.$ Hence, $x^2+y^2>0$ is true