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(1) I shift the origin to the point (h,k).

(2) Now the equation of the parabola in the new system becomes y[itex]^{2}[/itex] = 4ax.

(3) Now, we know that when we shift the origin without rotation of axes to a point (h,k) (wrt the old system), the locus in the old system is changed by replacing x by x-h and y by y-k. This gives us the equation of the locus in the new system.

Now, since the equation of the parabola in the new system is y[itex]^{2}[/itex]=4ax, then the equation of the parabola in the old system must be (y+k)[itex]^{2}[/itex] = 4a(x+h), as on replacing x by x-h and y by y-k in this equation, we get the equation of the parabola in the new system, which is y[itex]^{2}[/itex] = 4ax.

Please help in finding the fault in this derivation. Thanks!!!