Solution assignment 10 Equations with fractions

Given the hyperbola:

$H: y=\displaystyle\frac{1}{x}$

and the line:

$l: y=px+1$

For which value(s) of $p$ is the line tangent to the hyperbola?

In order to find the intersection points of $l$ with $H$ , we solve the following equation:

$\displaystyle\frac{1}{x}=px+1$

$px^2+x-1=0$

When $l$ is tangent to $H$ , this equation has two coinciding points and thus the discriminant $D=0$ :

$D=1+4p=0$

When $p$ has the value:

$p=-\displaystyle\frac{1}{4}$

the line is tangent to the hyperbola.