The distance from the center to each co-vertex is known as the semi-minor axis length.
Co-vertices are always located at $(h, k \pm b)$ or $(h \pm b, k)$ depending on the orientation of the ellipse.
For an ellipse with a horizontal major axis, co-vertices are aligned vertically and vice versa.
The coordinates of co-vertices can be found using the standard form equations $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ or $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ where $a > b$.
In real-world applications, understanding co-vertices helps in determining properties like eccentricity and focal points.
Review Questions
What are the coordinates of co-vertices for an ellipse centered at $(3, 4)$ with a vertical major axis and semi-minor length of 5?
If you know one co-vertex is at $(0, -6)$ and another at $(-8, -6)$, what's the center and orientation of this ellipse?
How do you distinguish between a vertex and a co-vertex in an equation of an ellipse?
Related terms
Major Axis: The longest diameter of an ellipse that passes through its center.
Minor Axis: The shortest diameter of an ellipse that passes through its center perpendicular to the major axis.
Eccentricity: A measure of how much an ellipse deviates from being circular; denoted as $e = \sqrt{1 - \frac{b^2}{a^2}}$ where $a$ is the semi-major axis length and $b$ is the semi-minor axis length.