Converting Parabola Equation from Axis Aligned Parametric to Explicit/Implicit Coordinate

1. The Parametric Equations for Axis Aligned Parabolas is given as the following
   
   \(x=B_1t + C_1,\hspace{.5cm}y=A_2t^2 + B_2t + C_2\)   ...(1) (For Parabolas having Directrix Parallel to \(X\)-Axis)
   
   \(x=A_1t^2 + B_1t + C_1,\hspace{.5cm}y=B_2t + C_2\)   ...(2) (For Parabolas having Directrix Parallel to \(Y\)-Axis)
   
   
2. Following are the steps to Convert the Non-Standard Parametric Equations to Explicit/Implicit 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 Explicit/Implicit Coordinate Equation.
   For Parabolas having Directrix Parallel to \(X\)-Axis this is done as follows
   
   From equation (1) we have,
   
   \(x=B_1t + C_1\hspace{.5cm}\Rightarrow \frac{x-C_1}{B_1}=t\)
   
   Also from equation (1) we have,
   
   \(y=A_2t^2 + B_2t + C_2\)
   
   \(\Rightarrow y=A_2\frac{{(x-C_1)}^2}{{B_1}^2} + B_2\frac{(x-C_1)}{B_1} + C_2\)   ...(3)
   
   \(\Rightarrow {B_1}^2y=A_2{(x-C_1)}^2 + B_1B_2(x-C_1) + {B_1}^2C_2\)
   
   \(\Rightarrow A_2x^2 + (B_1B_2 - 2A_2C_1)x -{B_1}^2y + A_2{C_1}^2 + {B_1}^2C_2 - B_1B_2C_1 =0\)   ...(4)
   
   The equation (3) and equation (4) above give the Explicit and Implicit Coordinate Equation respectively 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,
   
   \(y=B_2t + C_2\hspace{.5cm}\Rightarrow \frac{y-C_2}{B_2}=t\)
   
   Also from equation (2) we have,
   
   \(x=A_1t^2 + B_1t + C_1\)
   
   \(\Rightarrow x=A_1\frac{{(y-C_2)}^2}{{B_2}^2} + B_1\frac{(y-C_2)}{B_2} + C_1\)   ...(5)
   
   \(\Rightarrow {B_2}^2x=A_1{(y-C_2)}^2 + B_2B_1(y-C_2) + {B_2}^2C_1\)
   
   \(\Rightarrow A_1y^2 + (B_2B_1 - 2A_1C_2)y -{B_2}^2x + A_1{C_2}^2 + {B_2}^2C_1 - B_2B_1C_2 =0\)   ...(6)
   
   The equation (5) and equation (6) above give the Explicit and Implicit Coordinate Equation respectively for Parabolas having Directrix Parallel to \(Y\)-Axis.