Introduction to Conic Sections

By the end of this lesson, you will be able to identify the four conic sections (circle, parabola, ellipse, hyperbola), understand their geometric definition as cross-sections of a cone, and recognize each from its equation.

The Big Idea

Conic sections arise when a plane slices a double cone at different angles. This geometric origin — known since ancient Greece — gives rise to four distinct curves with fundamentally different shapes. Each conic has a precise algebraic equation, and identifying which conic you have from an equation is the first step in every problem.

The Four Conics

Circle: plane cuts cone parallel to the base. All points equidistant from a center.

Ellipse: plane cuts cone at a slight angle. Two focal points; sum of distances constant.

Parabola: plane cuts exactly parallel to one side of the cone. One focal point; equidistant from focus and directrix.

Hyperbola: plane cuts both nappes of the double cone. Two focal points; difference of distances constant.

Standard Forms (Quick Reference)

Conic	Standard Form
Circle	$(x-h)^2 + (y-k)^2 = r^2$
Parabola (vertical)	$(x-h)^2 = 4p(y-k)$
Ellipse (horizontal major)	$\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1$, $a > b$
Hyperbola (horizontal)	$\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1$

Identifying from General Form

The general second-degree equation is:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

For conics with axes parallel to the coordinate axes ($B = 0$):

Condition	Conic
$A = C$	Circle
$A \neq C$, same sign	Ellipse
$A$ or $C = 0$ (not both)	Parabola
$A$ and $C$ opposite signs	Hyperbola

Example 1 — Identifying from the Equation

(a) $4x^2 + 4y^2 - 8x + 12y - 3 = 0$: $A = C = 4$ → circle

(b) $3x^2 + y^2 - 6x + 4y - 2 = 0$: $A = 3$, $C = 1$, same sign, $A \neq C$ → ellipse

(c) $y^2 - 4x - 6y + 1 = 0$: $A = 0$, $C = 1$ (one is zero) → parabola