Question 1174826
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The solution in the post by @CPhill is incorrect.


It is incorrect TWO TIMES due to two reasons.


(1) The equation of the parabola is written in incorrect form.  


(2) in the solution, @CPhill makes an assumption that the tangent is along x-axis.

    The problem nowhere says it, so this assumption is WRONG and irrelevant to the problem.


To be correct, this assumption should not be made.


Google AI gives another solution to this problem (quite long) without making wrong assumptions.


See the link


https://www.google.com/search?q=find+the+equation+of+parabola+which+has+contact+of+third+order+with+the+conic+ax%5E2+%2B2hxy+%2B+by%5E2+%2B2gx+%2B2fy+%2Bc+%3D0+at+origin.&rlz=1C1CHBF_enUS1071US1071&oq=find+the+equation+of+parabola+which+has+contact+of+third+order+with+the+conic+ax%5E2+%2B2hxy+%2B+by%5E2+%2B2gx+%2B2fy+%2Bc+%3D0+at+origin.&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCTE5NTFqMGoxNagCCLACAfEFU-UGEVhaVo0&sourceid=chrome&ie=UTF-8