A hyperbola is a type of conic section that consists of two distinct branches. The co-vertices of a hyperbola are the points where the imaginary axis intersects the hyperbola’s transverse axis. These points are important in the geometry of hyperbolas as they help define the shape and spread of the curve.
To explain further, a hyperbola can be represented in standard form as:
For horizontal hyperbolas: (x - h)²/a² - (y - k)²/b² = 1 For vertical hyperbolas: (y - k)²/a² - (x - h)²/b² = 1
In these equations, (h, k) is the center of the hyperbola, ‘a’ is the distance from the center to each vertex, and ‘b’ is the distance from the center to each co-vertex.
The co-vertices can be found by taking the center (h, k) and moving ‘b’ units vertically (up and down for horizontal hyperbolas, or horizontally for vertical hyperbolas). Thus, the co-vertices are given by the coordinates:
- For a horizontal hyperbola: (h, k ± b)
- For a vertical hyperbola: (h ± b, k)
Understanding the co-vertices provides insight into how wide the hyperbola opens and the overall layout of the graph. They are critical points of reference for sketching and analyzing hyperbolas.