One-Parameter Family of Curves/Examples

Examples of One-Parameter Families of Curves

Concentric Circles

The equation:

$x^2 + y^2 = r^2$

is a one-parameter family of concentric circles whose centers are at the origin of a Cartesian plane and whose radii are the values of the parameter $r$.

Circles of Equal Radius with Centers along $x$-Axis

Consider the equation:

$(1): \quad \paren {x - h}^2 + y^2 = a^2$

where $a$ is constant.

$(1)$ defines a one-parameter family of circles of constant radius $a$ whose centers are on the $x$-axis of a Cartesian plane at $\tuple {h, 0}$ determined by values of the parameter $h$.