Converting Parabola Equation from Standard Coordinate to Standard Parametric

1. The Standard Coordinate Equations for Parabola having Vertex at \((x_v,y_v)\) and Signed Focal Length \(f\) is given as
   
   \({(x-x_v)}^2 = 4f(y-y_v)\)   (For Parabolas having Directrix Parallel to \(X\)-Axis)...(1)
   
   \({(y-y_v)}^2 = 4f(x-x_v)\)   (For Parabolas having Directrix Parallel to \(Y\)-Axis)...(2)
   
   
2. To Convert the Standard Coordinate Equations to Standard Parametric Equations the Quadratic Term of the Standard Coordinate Equation is set as a Real Number Parameter \(t\) and calculation are done as follows
   
   For Parabolas having Directrix Parallel to \(X\)-Axis let
   
   \(x-x_v=t\hspace{.5cm}\Rightarrow x=t+x_v\)
   
   Now, from equation (1) we have
   
   \({(x-x_v)}^2 = 4f(y-y_v)\)
   
   \(\Rightarrow t^2=4f(y-y_v)\)
   
   \(\Rightarrow \frac{t^2}{4f}=y-y_v\)
   
   \(\Rightarrow y=\frac{t^2}{4f} + y_v\)
   
   Hence, the Standard Parametric Equations for Parabolas having Directrix Parallel to \(X\)-Axis are given as
   
   \(x=t + x_v,\hspace{.5cm}y=\frac{t^2}{4f} + y_v\)   ...(3)
   
   Similarly, for Parabolas having Directrix Parallel to \(Y\)-Axis let
   
   \(y-y_v=t\hspace{.5cm}\Rightarrow y=t+y_v\)
   
   Now,from equation (2) we have
   
   \({(y-y_v)}^2 = 4f(x-x_v)\)
   
   \(\Rightarrow t^2=4f(x-x_v)\)
   
   \(\Rightarrow \frac{t^2}{4f}=x-x_v\)
   
   \(\Rightarrow x=\frac{t^2}{4f} + x_v\)
   
   Hence, the Standard Parametric Equations for Parabolas having Directrix Parallel to \(Y\)-Axis are given as
   
   \(y=t + y_v,\hspace{.5cm}x=\frac{t^2}{4f} + x_v\)   ...(4)
3. A more popular way to Convert the Standard Coordinate Equations to Standard Parametric Equations is by setting the Real Number Parameter \(t\) to the Quadratic Term of the Standard Coordinate Equation divided by 2 times the Signed Focal Length \(f\) and doing the calculations as follows
   
   For Parabolas having Directrix Parallel to \(X\)-Axis let
   
   \(\frac{x-x_v}{2f}=t\hspace{.5cm}\Rightarrow x-x_v=2ft\hspace{.5cm}\Rightarrow x=2ft+x_v\)
   
   Now, from equation (1) we have
   
   \({(x-x_v)}^2 = 4f(y-y_v)\)
   
   \(\Rightarrow 4f^2t^2 = 4f(y-y_v)\)
   
   \(\Rightarrow ft^2=y-y_v\)
   
   \(\Rightarrow y=ft^2 + y_v\)
   
   Hence, the Standard Parametric Equations for Parabolas having Directrix Parallel to \(X\)-Axis are given as
   
   \(x=2ft+x_v,\hspace{.5cm}y=ft^2 + y_v\)   ...(5)
   
   Similarly, for Parabolas having Directrix Parallel to \(Y\)-Axis let
   
   \(\frac{y-y_v}{2f}=t\hspace{.5cm}\Rightarrow y-y_v=2ft\hspace{.5cm}\Rightarrow y=2ft+y_v\)
   
   Now, from equation (2) we have
   
   \({(y-y_v)}^2 = 4f(x-x_v)\)
   
   \(\Rightarrow 4f^2t^2 = 4f(x-x_v)\)
   
   \(\Rightarrow ft^2=x-x_v\)
   
   \(\Rightarrow x=ft^2 + x_v\)
   
   Hence, the Standard Parametric Equations for Parabolas having Directrix Parallel to \(Y\)-Axis are given as
   
   \(y=2ft+y_v,\hspace{.5cm}x=ft^2 + x_v\)   ...(6)
   
   The equations (5) and (6) have the property that the Co-efficient of the Quadratic Term of Parameter \(t\) gives the Signed Focal Length \(f\) of the Parabola. Also the Co-efficient of the Linear Term of Parameter \(t\) is 2 times the Co-efficient of the Quadratic Term of Parameter \(t\) (i.e. 2 times the Signed Focal Length \(f\) of the Parabola.