Given a conic f(x,y) we can classify it studing the eigenvalues of its associated quadratic form. This quadratic form can be calculated as
q_f(x,y) =1/2 {x,y}cdot((f_{x x},f_{xy}),(f_{yx},f_{yy}))cdot {x,y}
Here f_{x_1,x_2} = (partial^2f)/(partial x_1partial x_2)
In this case we have
q_f(x,y) = {x,y}cdot((25,0),(0,4))cdot {x,y}
with eigenvalues {25,4} characterizing it as a parabola.
The standard form is
f_s(x,y) = {x-x_0,y-y_0}cdot((25,0),(0,4))cdot {x-x_0,y-y_0}+c_0
Doing f(x,y)-f_s(x,y)=0 and equating the coefficients we have
{(640 - c_0 - 25 x_0^2 - 4 y_0^2 = 0), (16 + 8 y_0 = 0), (50 (x_0-5) =0)
:}
solving for x_0,y_0,c_0 we get
f_s(x,y) = {x-5,y+2}cdot((25,0),(0,4))cdot {x-5,y+2}-1
