The parabola is given by the equation \(Y^{2}=X\) i.e. in parametrize form \(X=t^{2}\) and Y = t. The hyperbola is given by the equation XY = 1; i.e. in parametrize form \(X=t\) and \(Y=\frac{1}{t}\)

A parabola is a polynomial curve that can be parametrized by a polynomial function of the parameter (t). Whereas, a hyperbola require rational functions of ‘t’ that are not polynomials i.e. no polynomial parametrization is possible. Hence, instead of a polynomial curve, hyperbola is a rational curve. The highest degree term in a parabolic equation is \(Y^{2}\) i.e. the only factor Y is repeated twice. The highest degree term in a hyperbolic equation is XY which has the two factors i.e. X and Y.

Hyperbola and Parabola are conic sections generated by different methods. When a plane cuts through the cone (parallel to a cone’s side) a parabola is generated. The intersection of a cone and a plane (the plane is not parallel to the cone) results in Hyperbola.

The eccentricity of a parabola is one, whereas the eccentricity of a hyperbola is greater than one. The graph of a Hyperbola opens more widely than a parabola. Also, there are two curves in a hyperbola that are a mirror image of each other opening in opposing directions. Whereas, a parabola has just 1 curve.

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