You should have discovered that the above form only determines the parabola, hyperbola and ellipse. Write the equation for a circle using polar coordinates. Parametric equations are especially useful when describing a graph that is not a function. However, they can still be useful when the graph is a function. Now try to write a set of parametric equations to represent an hyperbola.
Many mathematical discoveries evolve over time.
The original meaning compared the base of the geometric figure constructed to the length of a given segment. The curves were of interest to the ancient geometers for use in trying to solve the construction problems of squaring a circle, doubling a square, and trisecting an angle. It took almost 1800 years for the mathematicians to see their practical use.