If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola,then what are the possible numberof places where the lines can intersec the hyperbola ?

I've played a bit with GeoGebra and it seems to me that I can get an infinite number of places on the hyperbola where two lines can intersect .

So I am misunderstanding somehow this question...

Can you guys give me some help to get on the right track ?

  • $\begingroup$ Each line can intersect the hyperbola in at most two points. $\endgroup$ – Lucian Dec 3 '15 at 15:57
  • $\begingroup$ That was simple T_T .Thank you sir. $\endgroup$ – Mr. Y Dec 3 '15 at 16:51

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.

  • $\begingroup$ It is possible for a line to intersect a hyperbola only once and not be tangent. Consider the hyperbola $y=1/x$ and the vertical line $x=1$. $\endgroup$ – Akiva Weinberger Dec 3 '15 at 16:37
  • $\begingroup$ @AkivaWeinberger. My bad error :(... Thank'you . I edit $\endgroup$ – Emilio Novati Dec 3 '15 at 16:56

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