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 $P=(x_P,y_P)$ the intersection point between the line and the hyperbola.
The hyperbola has equation $y=\frac{1}{x}$ so the derivative (without implicit differentiation) is $y'=\frac{-1}{x^2}$ and, at $P$ it is $y'(x_P)=\frac{-1}{x_P^2}$.
This is the slope of the tangent to the hyperbola at $P$, and if we want that it's orthogonal to the straight line $y=ax+\frac{1}{2}a-2$ we must have:
$\frac{-1}{x^2}=\frac{-1}{a} \Rightarrow a=x_P^2 $
Now, we want also that $P$ is a point of the straight line, se we must also have:
$\frac{1}{x_P}=ax_P+\frac{1}{2}a-2=x_P^3+\frac{1}{2}x_P^2-2 \Rightarrow x_P^4+\frac{1}{2}x_P^3-4x_P-2=0 \Rightarrow (2x_P-2)(x_P^3-2)=0$ 
Solving this equation you find  the possible values for $x_P$ and  for $a$.
