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

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

Solution: The equation x^2 - y^2 = c represents a family of hyperbolas with varying values of c.

Steps: 6

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.