I'm sorry that the title may sound like a mess, but that's the only way to describe the question.
The original question is in Portuguese and can be found here, it's question number 2.
Here's a translation of the question as I interpreted it:
Let
aandbbe real numbers, both not zero. The two lines that are perpendicular to $$ \dfrac x a + \dfrac y b = 1$$ and make triangles of area|ab|with the axes are:
And then it gives you a bunch of options and the correct answer is:
$$\dfrac x b - \dfrac y a = \sqrt{2} $$ and $$\dfrac y a - \dfrac x b = \sqrt{2}$$
I'm tired, and maybe I'm missing something simple, but I really don't know how to get to those answers, so any help is appreciated.
Edit: so in the interest of clarity, I'm going to draw what I think the question is asking
Where the red line is the given line, the blue and green lines are the answers, and the triangles are formed between blue-red-y-axis and green-red-x-axis
A perpendicular line to the first one has an equation which can be expressed as: $$bx-ay=c \tag{1}$$ The intersection of that line with the axes will produce two points:
$A=(0, -\frac{c}{a})$ and $B=(\frac{c}{b},0)$
The triangle AOB is a right-angled one, and its area is: $$[AOB]=\frac{c^2}{2|ab|}.$$ Since the area must be equal to $|ab|$, we conclude that: $$c=\pm \sqrt{2}|ab| \tag{2}.$$ Substituting (2) in (1) we get both answers.