Conic sections
Also known as: conics
Conic sections are the four curves formed when a plane slices through a double cone: the circle, ellipse, parabola, and hyperbola. Each has a standard equation that reveals its center, vertices, and shape.
Conic sections are the curves produced by cutting a double cone with a flat plane at different angles. Slice perpendicular to the cone's axis and you get a circle; tilt the plane and you get an ellipse; make it parallel to the cone's side and you get a parabola; cut steeply through both halves of the cone and you get a hyperbola.
Each conic has a standard equation. A circle with center (h, k) and radius r is (x − h)² + (y − k)² = r². An ellipse centered at (h, k) is (x − h)²/a² + (y − k)²/b² = 1 — the sum of two squared terms set equal to 1. A hyperbola looks similar but subtracts: (x − h)²/a² − (y − k)²/b² = 1. A parabola has only one squared variable, such as y = a(x − h)² + k.
Recognizing a conic from its equation is mostly a matter of inspecting the squared terms. Both variables squared with equal coefficients and addition: circle. Both squared with different positive coefficients: ellipse. One squared term subtracted from the other: hyperbola. Only one variable squared: parabola. From the standard form you can then read off the center, radius, vertices, or direction of opening.
On the ACT, conic sections appear in the plane geometry and coordinate geometry questions — most often circles, where you must extract the center and radius from the equation or write the equation from a graph, along with parabolas in vertex form. Being able to classify an equation at a glance and read off its key features is the skill the exam rewards.
Key takeaways
- The four conic sections — circle, ellipse, parabola, hyperbola — come from slicing a double cone at different angles.
- A circle's standard equation is (x − h)² + (y − k)² = r², with center (h, k) and radius r.
- Ellipses add two squared terms set equal to 1; hyperbolas subtract them; parabolas square only one variable.
- Classify a conic by inspecting its squared terms and their signs.
- ACT questions focus on circles and parabolas: identify the center, radius, or vertex from the equation.