Converting Parabola Equation from Standard Parametric to Standard Coordinate

1. The Standard Parametric Equations for Parabola having Vertex at \((x_v,y_v)\) are given as
   
   \(x=B_1t + x_v,\hspace{.5cm}y=A_2t^2 + y_v\)   (For Parabolas having Directrix Parallel to \(X\)-Axis)...(1)
   
   \(x=A_1t^2 + x_v,\hspace{.5cm}y=B_2t + y_v\)   (For Parabolas having Directrix Parallel to \(Y\)-Axis)...(2)
   
   
2. Following are the steps to Convert the Standard Parametric Equations to Standard Coordinate Equation
   1. Find the value of Parameter \(t\) from the Parametric Equation having the Linear Term of Parameter \(t\).
   2. Plug in the value of Parameter \(t\) thus obtained in the Parametric Equation having the Quadratic Term of Parameter \(t\) and re-arrange the equation to form the Standard Coordinate Equation.
   For Parabolas having Directrix Parallel to \(X\)-Axis this is done as follows
   
   From equation (1) we have,
   
   \(x=B_1t + x_v\hspace{.5cm}\Rightarrow \frac{x-x_v}{B_1}=t\)
   
   Also from equation (1) we have,
   
   \(y=A_2t^2 + y_v\)
   
   \(\Rightarrow \frac{y-y_v}{A_2}=t^2\)
   
   \(\Rightarrow \frac{y-y_v}{A_2}=\frac{{(x-x_v)}^2}{{B_1}^2}\)
   
   \(\Rightarrow \frac{{B_1}^2}{A_2}(y-y_v)={(x-x_v)}^2\)...(3)
   
   The equation (3) above gives the Standard Coordinate Equation for Parabolas having Directrix Parallel to \(X\)-Axis.
   
   Similarly, for Parabolas having Directrix Parallel to \(Y\)-Axis this is done as follows
   
   From equation (2) we have,
   
   \(y=B_2t + y_v\hspace{.5cm}\Rightarrow \frac{y-y_v}{B_2}=t\)
   
   Also from equation (2) we have,
   
   \(x=A_1t^2 + x_v\)
   
   \(\Rightarrow \frac{x-x_v}{A_1}=t^2\)
   
   \(\Rightarrow \frac{x-x_v}{A_1}=\frac{{(y-y_v)}^2}{{B_2}^2}\)
   
   \(\Rightarrow \frac{{B_2}^2}{A_1}(x-x_v)={(y-y_v)}^2\)...(4)
   
   The equation (4) above gives the Standard Coordinate Equation for Parabolas having Directrix Parallel to \(Y\)-Axis.