1. The problem statement, all variables and given/known data We have the hyperbola, the focal stuff of which is on the Abscissa axis. $$x^2 - 2y^2 = 4 $$, and we have a line $$3x - 4y = 2$$, and we need to understand if this two crazy stuff will intersect, or be tangent, or nothing like the previous one. 2. Relevant equations I don't clearly understand what is means "Relevant Equations". 3. The attempt at a solution Well, first time I have made a mistake with a signs. In general, solution is next - we trying to find the points that belongs to each crazy stuff simultaneously, it's means this points a roots of the two equations in the same time. I have made another few attempts, and in the one attempt, I have tried to change the order of the equations, and try to find a roots of Hyperbola, and then put it to the Line geometric-algebraic model. Despite, and at least, when I have typing a request to this forum, I have found the true solution that fits with a book' answer. so the question why despite of my few mistakes some wrong solutions satisfying the one of the equations in the each case?! Thanx. ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// Ooooops. My lot of pardons, I have realized why some wrong solution is satisfying some particular equations, that because in the first attempt $$x \in (- \infty , + \infty)$$, same as $$y \in (- \infty , + \infty)$$, and I have made mistake only in the equation of the Hyperbola, so if we gonna put any number in the equation of the Line, and then with no mistake find the another coord. variable - we will obtain the right answer. And in the second attempt, despite it's attempt was in the first coord. angle, due to avoiding less than zero values inside of the square root, we can say the same as we said about the Line equation. :) Me happy ^^.