My assignment question is to use Lagrange multipliers to find the points of minimum distance from the origin to the curve $4x^2-10xy+4y^2=36$.
Taking my main function for the purpose of finding critical points as the function of the distance squared, i.e. $f(x,y)=x^2+y^2$ constrained by $g(x,y)=4x^2-10xy+4y^2$, I get the following system of equations to solve:
$f_x=\lambda g_x, \; f_y=\lambda g_y, \; 4x^2-10xy+4y^2=36$
Because the first two equations are symmetrical, $2x=\lambda (8x-10y)$ and $2y = \lambda (8y-10x)$ simplify to $y^2=x^2$ or $y=\pm x$. But plugging the latter into $4x^2-10xy+4y^2=36$ gives me the square root of a negative number. This would appear to show that points of minimum or maximum distance do not exist, but graphing the problem shows that the curve is a set of two hyperbolas which are definitely closest to 0 when $y=\pm x$.
I would really appreciate if someone could explain to me what I might be doing wrong here. I do specifically need to use the Lagrange multipliers method to solve this problem, as per the requirements of the assignment.