๐ Topic Summary
A hyperbola is a type of conic section defined as the set of all points where the difference of the distances to two fixed points (foci) is constant. The standard form of a hyperbola depends on whether it opens horizontally or vertically. Understanding the standard form is key to identifying the hyperbola's center, vertices, and asymptotes.
For a hyperbola centered at $(h, k)$ that opens horizontally, the standard form is: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$. If it opens vertically, the standard form is: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$. Here, $a$ is the distance from the center to each vertex, and $b$ is related to the distance to the co-vertices. The relationship between $a$, $b$, and the distance from the center to each focus, $c$, is given by $c^2 = a^2 + b^2$.
๐ค Part A: Vocabulary
Match the following terms with their definitions:
- Hyperbola
- Center
- Foci
- Vertices
- Asymptotes
Definitions:
- Lines that the hyperbola approaches but never touches.
- The points on the hyperbola closest to the center.
- The midpoint between the two vertices.
- The set of all points where the difference of the distances to two fixed points is constant.
- The two fixed points that define the hyperbola.
Write your answers here (e.g., 1-D, 2-C, etc.):
โ๏ธ Part B: Fill in the Blanks
A hyperbola is defined by its ________, ________, and ________. The standard form of a hyperbola centered at (h,k) that opens horizontally is $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$, where 'a' represents the distance from the center to the ________, and the relationship between a, b, and c (distance to foci) is $c^2 = ________ + ________$.
๐ค Part C: Critical Thinking
Explain how changing the values of 'a' and 'b' in the standard form equation of a hyperbola affects its shape and orientation.