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The following are some examples of equations of parabolas in x-y plane.

the vertex is at origin and a gives the distance focus from vertex.
 x^{2} = 4 a y \\ 
 y^{2} = 4 a x \\ 

passes through origin vertex at A/2B 
y = A x - B x^{2} \\

another one
                x = A y - B y^{2} \\ 

vertex at (x1,y1) and focal length =  a

           (x-x1)^{2} = 4 a (y-y1)

well, the following
         Ax^{2} + Bxy + C y^{2} + Wx + U y + V = 0
represents  parabola  if B² = 4 AC and more conditions ....
1 5 1
Y^2=4ax  and  x^2=4ay is the equation of parabola
1 5 1
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this should never be marked as best... face the truth......
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don't cry....
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