The question is to find the equations of the lines (plural) that are parallel to the graph of $y=-x+6$ and perpendicular to the graph of $y=1/x$. I understand that to be parallel to the line, the slope would be $-1$. The derivative of the curve is $-1/x^2$. The tangent would be opposite reciprocal...so $x^2$ Now I'm stuck. I can easily calculate the points of intersection of the line and curve as $(5.828, 0.172)$ and $(0.172, 5.828)$, but not clear how that helps.
Could I just choose a random point, say $(0,0)$ and calculate a line parallel to the given line to be $y=-x$. That line is parallel to the given line. So are an infinite number of others.
Then to find a line perpendicular to the curve I could use $(1,1)$ (random choice) and get $y'=1/1$, so $y=x$.
These lines seem to answer the question but seems too arbitrary.