Final answer:
The identification of the elements of a vertical hyperbola includes locating the center, vertices, foci, as well as describing the orientation and length of both the transverse and conjugate axes.
Step-by-step explanation:
The question is asking for the identification of various elements associated with vertical hyperbolas. To address the student's question, we must assume there is a general equation provided for a vertical hyperbola centered at the origin, which would be written as −\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, where the center is at (h, k).
- Center: The coordinates of the center are (h, k).
- Vertices: For a vertical hyperbola, the vertices are located at (h, k ± a); vertically upward and downward in the coordinate system.
- Foci: They lie along the transverse axis at a distance c from the center, where c^2 = a^2 + b^2. The coordinates of the foci for a vertical hyperbola are (h, k ± c).
- Transverse Axis: This is the line segment through the vertices, extending from (h, k - a) to (h, k + a), which is vertically oriented.
- Conjugate Axis: This is the line segment that is perpendicular to the transverse axis and has a length of 2b, extending from (h - b, k) to (h + b, k), which is horizontally oriented.