mail  mail@stemandmusic.in

    

call  +91-9818088802

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.