The general equation of a straight line is linear: $ax+by+c=$ and the general equation of a hyperbola (but it's the same for all conics) is a second degree equation: $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ so, finding $y$ ( or $x$ ) from the first equation and substituting in the second, we have a second degree equation that has, at most, two real solutions.
The possible cases are:
1) two real distinct solutions: the line is secant at two points
2) two coincident solution: the line is tangent
3) no real solution : the line is e''external''
4) the equation reduce to a first degree equation so we have one real solution and we say that the other solution goes to infinity.
So, for two lines that are not tangent we can have:
no common point with the conic if the two lines are external,
$2$ points if one is secant at two pints and the other is external or if both have one real common point with the conic an the other at infinity,
$3$ common points if one is secant at two points and the other at one point,
$4$ common points if the two lines are both secants at two points.