A co-vertex of an ellipse is one of the endpoints of the minor axis. These points lie on the line perpendicular to the major axis through the center of the ellipse.
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For an ellipse centered at $(h, k)$ with semi-major axis $a$ and semi-minor axis $b$, the co-vertices are located at $(h, k \pm b)$ for a vertical minor axis or $(h \pm b, k)$ for a horizontal minor axis.
The distance from the center to each co-vertex is equal to the length of the semi-minor axis ($b$).
In standard form, if the equation of an ellipse is given by $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, then $b$ represents half the length of the minor axis.
Co-vertices help in sketching and understanding the dimensions and orientation of an ellipse.
The coordinates of co-vertices change depending on whether $a > b$ (horizontal major axis) or $b > a$ (vertical major axis).
Review Questions
Where are the co-vertices located for an ellipse centered at $(3, -2)$ with a horizontal semi-major axis of length 5 and a semi-minor axis length of 3?
How do you determine whether to add or subtract $b$ to/from $k$ or $h$ when finding co-vertices?
What role do co-vertices play in determining the shape and orientation of an ellipse?
Related terms
Ellipse: A set of all points in a plane such that the sum of their distances from two fixed points (foci) is constant.
Major Axis: The longest diameter of an ellipse that passes through its center and both foci.
Minor Axis: The shortest diameter of an ellipse that passes through its center and is perpendicular to its major axis.