I'm not able to follow the logic of your method, but it does not seem to give a valid answer. The original family is g(x, y) = x^{3}/y^{2} = a; if your answer f(x, y) = y^{2}+x = c is correct then you should find ∇f.∇g = 0. ∇g = (3x^{2}/y^{2}, -2x^{3}/y^{3}); ∇f = (1, 2y); ∇f.∇g = 3x^{2}/y^{2} - 4x^{3}/y^{2}.
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling We Value Civility
• Positive and compassionate attitudes
• Patience while debating We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving