Final answer:
To identify elements of a hyperbola, the center is midway between vertices, the foci are interior points further from the center, and asymptotes are equations represented by lines that the hyperbola approaches but does not touch.
Step-by-step explanation:
The student is asking how to identify the center, vertices, foci, and equations of the asymptotes of a hyperbola. The center of a hyperbola is the midpoint between the two vertices, which are the points on each arm of the hyperbola where it curves the most. The foci are two fixed points located inside each branch of the hyperbola, and they are further from the center than the vertices.
The equations of the asymptotes depend on the distances from the center to the vertices (a) and foci (c). They are straight lines that the curve of the hyperbola approaches but never touches. The general equations for the asymptotes of the hyperbola centered at the origin with horizontal transverse axis are y = ±(b/a)x, where 'b' is the distance from the center to a vertex along the y-axis.
When we look at an ellipse, each semi-major axis is denoted by 'a,' and the distance 2a is called the major axis of the ellipse. The two foci (representing the special points) are located along the major axis, with one focus usually being the central celestial body, such as the Sun in the context of planetary orbits. The other focus is empty space, but both are equidistant from the center of the ellipse.