AP® Precalculus · Unit 4 · Topic 4.6
Ellipses
An ellipse is a circle that's been stretched by different amounts horizontally and vertically, and its equation (x-h)2/a2 + (y-k)2/b2 = 1 records exactly how much by giving each direction its own radius.
Take a circle and pull it wider or squish it flatter in one direction only, and you get an ellipse. A circle treats every direction the same: one radius covers horizontal, vertical, everything. An ellipse breaks that symmetry. It still has a center, but now it needs two radii — one for how far it reaches left and right, another for how far it reaches up and down.
The standard equation (x-h)2/a2 + (y-k)2/b2 = 1 keeps the center at (h, k), just like a circle. The difference is underneath: a2 sits under the x-term and controls the horizontal radius, b2 sits under the y-term and controls the vertical radius. Plug in y = k and the equation forces (x-h)2 = a2, so x lands exactly a units left and right of center — that's the horizontal radius. Set x = h instead, and (y-k)2 = b2 puts the vertex b units above and below center. Same structure as a circle, just two dials instead of one.
This is why a circle is the special case where a = b: both radii agree, the equation collapses to (x-h)2 + (y-k)2 = a2, and you're back to one radius in every direction. Ellipses fit naturally next to circles, parabolas, and other parameter-controlled curves in this unit — same idea of a center plus numbers that stretch a basic shape, just applied with two independent stretch factors instead of one.
Key terms
- Center
- The point (h, k) that both radii are measured from in the ellipse equation.
- Horizontal radius (a)
- The distance from center to the ellipse's left and right vertices, found from a2 under the x-term.
- Vertical radius (b)
- The distance from center to the ellipse's top and bottom vertices, found from b2 under the y-term.
- Vertex
- A point where the ellipse crosses its horizontal or vertical axis through the center.
Watch it explained
Check yourself
End of topic.