What is the angle between the asymptotes of the hyperbola $5x^2-2\sqrt 7 xy-y^2-2x+1=0$? 
What is the angle between the asymptotes of this hyperbola?
  $$5x^2-2\sqrt 7 xy-y^2-2x+1=0$$

I used $S+\lambda=0$ and used straight line condition to find combined equation to asymptotes. Then how to find angle between them?

Rearranging lab's answer ...  
Omit the lower-degree terms of $5x^2-2\sqrt 7 xy-y^2-2x+1=0$ to get $5x^2-2\sqrt 7 xy-y^2=0$.  This is the equation of two lines which are parallel to your asymptotes.  So find the angle between these two lines.  
Factor:
$$
5x^2-2\sqrt 7 xy-y^2 =
-\frac{\left( 2\,y\sqrt {3}-\sqrt {7}y+5\,x \right)  \left( 2\,y\sqrt 
{3}+\sqrt {7}y-5\,x \right) }{5}
$$ 
Solving and rationalizing, the lines are
$$
y=(2\sqrt{3}-\sqrt{7})x,\quad y=(-2\sqrt{3}-\sqrt{7})x
$$
Use the addition formula for tangents as lab suggests.
