ACT® · Math: Geometry — Coordinate Geometry and Conic Sections
The Equation of an Ellipse
An ellipse is a stretched circle: x2/a2 + y2/b2 = 1, where a2 and b2 tell you how far it reaches along each axis. The bigger denominator always marks the longer, major axis.
An ellipse is what you get when you pull a loop of string taut around two fixed pins and trace around them. The string length is fixed, so the sum of the distances from any point on the curve to the two pins (the foci) never changes. That constant-sum rule is what bends a circle into an oval: near the ends of the long axis, the point is far from one pin and close to the other; near the short axis, it's a more even split, but the total stays the same.
The standard equation x2/a2 + y2/b2 = 1 encodes this shape with two numbers. Setting y = 0 gives x = plus or minus a, so the ellipse crosses the x-axis at distance a from center. Setting x = 0 gives y = plus or minus b. Whichever of a2, b2 is larger tells you which axis is longer, the major axis, with the other being the minor axis. The center sits at the origin in this basic form, and the vertices are the endpoints of the major axis.
This sits right after circles in coordinate geometry because the equation looks almost identical, just with two different denominators instead of one shared radius squared. It leads naturally into the hyperbola, where a minus sign between the terms replaces the plus sign, and into what happens when a and b become equal: the ellipse becomes a circle, and when the foci merge into one point, the string loop traces a perfect circle of radius a.
Key terms
- Major axis
- The longer axis of the ellipse, with length twice the square root of the larger denominator.
- Minor axis
- The shorter axis of the ellipse, with length twice the square root of the smaller denominator.
- Vertices
- The two endpoints of the major axis, the points on the ellipse farthest from the center.
- Foci
- The two fixed points inside the ellipse whose distances from any point on the curve always add to the same total.
- Center
- The midpoint between the two foci and the point of symmetry of the ellipse.
Watch it explained
- How it worksHow do the denominators in x2/a2 + y2/b2 = 1 set the two axis lengths of an ellipse?
- Worked exampleFind the center, the vertices and both axis lengths of x2/25 + y2/9 = 1.
- Common mistakeWhy isn't the larger denominator always the one sitting under the x term?
- Going furtherWhat happens to an ellipse as its two foci slide together into a single point?
Check yourself
End of topic.