Implicit Functions and Conic Sections

Curves defined by equations, not formulas

An implicitly defined function is the set of points $(x,y)$ satisfying an equation such as $x^2+y^2-6x+4y=12$ — no one variable is isolated. To test whether a point lies on the curve, substitute both coordinates; to understand the shape, complete the square in each variable.

The three conics in standard form

• Parabola: $(x-h)^2=4p(y-k)$ or $(y-k)^2=4p(x-h)$ — only one variable is squared. Vertex $(h,k)$, focus a distance $|p|$ from it.
• Ellipse: $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ — a sum equal to 1. Foci lie on the major axis with $c^2=a^2-b^2$.
• Hyperbola: $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ — a difference equal to 1. Opens toward the positive term; $c^2=a^2+b^2$; asymptotes pass through the center with slopes $\pm\frac{b}{a}$ (for the horizontal orientation).

Fast classification of $Ax^2+Cy^2+Dx+Ey+F=0$: one squared term $\Rightarrow$ parabola; $A,C$ same sign $\Rightarrow$ ellipse (circle if $A=C$); opposite signs $\Rightarrow$ hyperbola.