Implicit differentiation gives too many stationary points

[nomadreid]

If, with y a function of x, I have the equation x²-5xy+3y² = 7, then by implicit differentiation, I get that dy/dx = (2x-5y)/(5x-6y). This equals zero everywhere on the straight line y=(2/5)x except at the origin. This would seem to indicate stationary points everywhere on that line, which is hard to imagine, and anyway the graph of this equation seems to be a hyperbola, having no stationary points. Obviously I am missing something very basic here. Any help would be appreciated.

 

[Mute]

The line y = 2x/5 does not intersect your hyperbola, so there are no values of x or y that satisfy both the original equation and the stationary point equation. Hence, no stationary points. (Looking at the plot it might not be obvious that the curves don't intersect, so you can also plug y = 2x/5 back into your implicit equation and solve for x, which gives you imaginary solutions, confirming that the curves don't intersect).

 

[nomadreid]

Ah. I should have spotted that. Thanks very much, Mute!