Question: Two rays in the first quadrant: $$x +y = |a|$$ $$ax - y = 1$$ intersect each other in the interval $a \in (a_0, \infty)$, the what is the value of $a_0$?
I don't even understand where to begin this question. I tried plotting a rough sketch of the lines by assuming a random value for $a$, but that didn't get me anywhere. Please help!
We may suppose that $a\not =0$.
The line $x+y=|a|$ passes through two points $(0,|a|),(|a|,0)$. And the line $ax-y=1\iff y=ax-1$ passes through the point $(0,-1)$ with the slope $a$.
Since the two lines intersect in the first quadrant, the slope $a$ of the second line has to be larger than $a_0$ such that $y=a_0x-1$ passes through the point $(|a_0|,0)$. Hence we have $$0=a_0|a_0|-1\iff a_0|a_0|=1.$$ If $a_0\gt 0$, then $(a_0)^2=1$ leads $a_0=1$.
If $a_0\lt 0$, then $-(a_0)^2=1$ has no real solution.
Therefore, we have $a_0=1$.