The above formulae can also be used to find the Co-efficients of the General Parametric Equations of Parabola, when the Implicit Coordinate Equation of Parabola is given.
The following gives the algorithm for the same
From equation (1) given above we see that the Co-efficient of \(x^2\hspace{.3cm}A={A_2}^2\). Therefore the value of \(A_2\) can be one of \(\pm \sqrt{A}\).
From equation (3) given above we see that the Co-efficient of \(y^2\hspace{.3cm}C={A_1}^2\). Therefore the value of \(A_1\) can be one of \(\pm \sqrt{C}\).
From equation (2) given above we see that the Co-efficient of \(xy\hspace{.3cm}B=-2A_1A_2\). Therefore if the value of \(B < 0 \) then value of both \(A_1\) and \(A_2\) are either Positive or Negative. However, if the value of \(B > 0 \) then value of one of \(A_1\) and \(A_2\) is Negative and the other is Positive.
Since we do not know what are the actual signs for \(A_1\) and \(A_2\), to do further calculations for the time being we assign arbitrary signage to these Co-efficients based on value of \(B\).
Now, to find the value of Co-efficients Variables \(B_1\), \(B_2\), \(C_1\) and \(C_2\) we need 4 equations. However we only have 3 equations (equation (4), (5) and (6)), which provide the value of these Variables.
Therefore we have to fix value of one of Co-efficients \(B_1\) or \(B_2\) as 0. The values of the remaining Co-efficient Variables are calculated as follows
When value of \(B_2\) is fixed as 0, the value of \(B_1\), \(C_1\) and \(C_2\) are calculated as follows
If the value of \({B_1}^2 > 0\) in equation (9) given above, then our assignment of sign for \(A_1\) and \(A_2\) in the previous was correct.
However, if the value of \({B_1}^2 < 0\) in equation (9) given above, then we must multiply value of both \(A_1\) and \(A_2\) with -1 (i.e invert the sign of \(A_1\) and \(A_2\)) to provide them the correct sign.
Also, the value of \(B_1\) can be found out as \(\sqrt{|{B_1}^2|}\).
If the value of \({B_2}^2 > 0\) in equation (14) given above, then our assignment of sign for \(A_1\) and \(A_2\) in the previous was correct.
However, if the value of \({B_2}^2 < 0\) in equation (14) given above, then we must multiply value of both \(A_1\) and \(A_2\) with -1 (i.e invert the sign of \(A_1\) and \(A_2\)) to provide them the correct sign.
Also, the value of \(B_2\) can be found out as \(\sqrt{|{B_2}^2|}\).
If both the Co-efficients of \(x^2\) (i.e. value of \(A\)) and \(y^2\) (i.e. value of \(C\)) in the Implicit Coordinate Equation of Parabola are Non-Zero, then 2 Sets of Parametric Equations can be found out for such Parabola (one by setting the value of \(B_1\) to 0 and other one by setting the value of \(B_2\) to 0). If the Co-efficient of \(x^2\) (i.e. value of \(A\)) in the Implicit Coordinate Equation of Parabola is Zero, then only one Set of Parametric Equations can be found out for such Parabola which will have \(B_1\) set to 0 (since the Co-efficient \(A_2\) will be equal to 0 and hence \(B_2\) cannot be zero) .
Similarly, if the Co-efficient of \(y^2\) (i.e. value of \(C\)) in the Implicit Coordinate Equation of Parabola is Zero, then only one Set of Parametric Equations can be found out for such Parabola which will have \(B_2\) set to 0 (since the Co-efficient \(A_1\) will be equal to 0 and hence \(B_1\) cannot be zero).