Question

1. Define a hyperbola in terms of its foci.2. What can we conclude about a hyperbola if it’s asymptotes intersect at the origin?3. What must be true of the foci of a hyperbola?4. If the transverse axis of a hyperbola is vertical, what do we know about the graph? 5. Where must the center of a hyperbola be relative to its foci?

Answer 1

1. Hyperbola is the set of all points such that the difference between any point (x, y) and the foci is a constant.

2. The asymptotes of a hyperbola will from a shape of x and the intersection of these asymptotes is also the vertex/center of the hyperbola.

So if the asymptotes intersect at the origin, the vertex/center of the hyperbola lies at the origin.

3. The foci of a hyperbola lie inside the hyperbola itself and located along the transverse axis.

4. The transverse axis of a hyperbola defines the orientation. It's either vertical, horizontal or slanted.

If the transverse axis is vertical, then the hyperbola is also in vertical orientation.

5. The center of hyperbola is located along the transverse axis. Since foci are also located along this axis, the center and the foci are collinear or they lie on the same line.