Determine the family of curves represented by the equation x^2 - y^2 = c, where

Determine the family of curves represented by the equation x^2 - y^2 = c, where c is a constant.

Determine the family of curves represented by the equation x^2 - y^2 = c, where c is a constant.

Step 1: Understand the equation x^2 - y^2 = c. This is a mathematical equation involving x and y.
Step 2: Recognize that c is a constant. This means it can take different values (like 1, 2, -1, etc.).
Step 3: Rearrange the equation to see its form: y^2 = x^2 - c. This helps in identifying the type of curve.
Step 4: Notice that the equation resembles the standard form of a hyperbola, which is generally written as (x^2/a^2) - (y^2/b^2) = 1.
Step 5: Realize that as c changes, the shape and position of the hyperbola change, but they all belong to the same family of curves.
Step 6: Conclude that the equation x^2 - y^2 = c represents a family of hyperbolas, where each hyperbola corresponds to a different value of c.

Hyperbolas – The equation x^2 - y^2 = c describes hyperbolas that open along the x-axis, with the value of c determining the specific hyperbola.
Family of Curves – The term 'family of curves' refers to a set of curves that share a common equation but differ based on the parameter (in this case, c).