Find all the parameters and such that the line $y = ax + \frac{1}{2}a - 2$ intersects the hyperbola $xy = 1$ at right angles in at least one point . Problem:
Find all the parameters and such that the line
$y = ax + \frac{1}{2}a - 2$ intersects the hyperbola $xy = 1$ at right angles
in at least one point.
My work:
We try to find tangent to hyperbola such that tangent line intersects given line at rigth angle. We know that two line $y_1=ax+b$ and $y_2=cx+d$ intersects at right angle iff $a=-\frac{1}{c}$.
 So we need to find tangent to hyperbola. Let $t...y=kx+l$ be our tangent line. To find that first we will need $y'$ and we will find it by implicit differentiation.
$$xy=1$$
$$y+xy'=0 \iff y'=\frac{-y}{x}$$
$$k=-\frac{y}{x}$$
But I haven't idea how to finish my work. 

Let $ (x_1, y_1) $be the point of intersection note that 
$ y_1 = a x_1 - 1/2 a - 2 $ and $ y_1 x_1 = 1 $. the slope at the point $ (x_1, y_1) $ of the line is $ a $ for the hyperbola $ y' = \frac{-1}{(x_1)^2}$ hence $ a = (x_1)^2$ three equations with the variables $ a, x_1,  y_1$
