Minimizing distance from origin on a curve.

Solution 1:

Not sure about your method, but another way is to look at it as minimizing $x^2+y^2$ (Technically the square root of that, but distance is minimized when distance squared is minimized and it's easier to work with the latter) for points on your curve.

Now, you can view the curve as a quadratic equation in y:

$$2y^2+(4x)y+(5x^2-6)=0$$ and use either completing the square or the equivalent quadratic formula with $a=2, b=4x, c=5x^2-6$ to find the solutions for $y$ in terms of x. Note there will be two of them as the graph is not a function.

Then you can just plug those values of $y$ into $x^2+y^2$ and you are reduced to a function in a single variable $x$ which you can minimize by 1 variable means, taking derivative, finding critical points, etc.

I'm not sure how feasible/calculable that route is, but it's the other main way you can do problems of "Optimize a function subject to a constraint": Solve the constraint for one variable and plug into function to reduce it to a lower variable function