If the line at infinity is a secant line of a conic then the conic is a hyperbola?

I assume you are experimenting in Geogebra to get an intuition for this concept, and I'll answer from that point of view.

What you're currently doing is making $A,B$ approach the same point at infinity. As a result you are constructing a conic that meets the line at infinity at a single point, and thus has the line at infinity as a tangent, i.e. the conic is a parabola.

Instead, draw two lines $a,b$ intersecting at the origin, and then create a third line $c$. Let points $A,B$ be the intersections of $c$ with $a,b$. Then create a conic that goes through the points $A,B,C,D,E$ and move $c$ towards infinity. You'll tend towards a hyperbola. (It's more interesting if you choose $C,D,E$ so that your conic starts off as an ellipse.). Using this construction you'll ensure that $A,B$ approach different points at infinity, and you'll get a conic for which the line at infinity is a true secant.

Imagine three dimensional Euclidean space with the origin at $\,O\,$ and a plane $\,P\,$ not containing $\,O.\,$ The projective plane associated with the space is all of the lines that pass through $\,O.\,$ Each of these lines intersects the plane $\,P\,$ except those that lie in the plane $\,Q\,$ that passes through the origin $\,O\,$ and is parallel to $\,P.\,$ Thus, the points in $\,P\,$ represent ordinary projective points. Pick a circle $\,L\,$ with center $\,O\,$ that lies in $\,Q.\,$ Pairs of antipodal points in $\,L\,$ are the points in the line at infinity.

Now pick a double cone $\,D\,$ with center at $\,O.\,$ The intersection of $\,D\,$ with plane $\,P\,$ is a conic section $\,C.\,$ The number of points at infinity of $\,D\,$ is at most two. The case with two points at infinity are those where $\,C\,$ is a hyperbola because two lines of $\,D\,$ pass through $\,O\,$ and also intersect $\,L\,$ in two points at infinity. Note that a Euclidean hyperbola has two connected open real components because they omit the two points at infinity that connect them. Similarly a Euclidean parabola has one open real component and is missing one point at infinity.