Family of Curves/Examples

Examples of Families of Curves

Circles with Centers along $x$-Axis

Consider the equation:

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

where $x, y, a, h \in \R$.

$(1)$ defines a family of circles:

whose radii are determined by the parameter $a$

whose centers are on the $x$-axis of a Cartesian plane at $\tuple {h, 0}$ determined by values of the parameter $h$.

Circles in Plane

Consider the equation:

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

$(1)$ defines a family of circles:

whose radii are determined by the parameter $a$

whose centers are at $\tuple {h, k}$ on Cartesian plane, determined by values of the parameters $h$ and $k$.

Hence $(1)$ represents the family of all circles in the plane.