A double in this example problem

[Muthumanimaran]

I am currently self-studying Mathematical Methods for Studying by Weber, Arfken and Harris. In the chapter Matrix Eigenvalue problems, I'm stuck in a particular step in a particular problem. Please look at the Image attached. The expression which is boxed, $$\frac{F_x}{F_y}\neq\frac{x}{y}$$ in the textbook as we can see, it is said that the force will not be directed toward the minimum at $x=0$ & $y=0$. Actually this is the step I don't understand, how the expression came up here. I am trying to visualize this problem physically, but I am unable to come up with this expression. Please give me some hint or information if possible.

 

[MisterX]

If the ratios are equal it means that the angle of the force can be the same as the angle from the origin to the position (x,y).
$$cot^{-1}(\frac{x}{y}) = \theta_{X} \;\;\;\;cot^{-1}(\frac{F_x}{F_y}) = \theta_{F} $$
$$\frac{x}{y} =\frac{F_x}{F_y}\; \text{implies}\;\theta_{X}=\theta_{F}$$
That is not the case for the given problem, where most of the places the angles are different. Another way of writing this could be
$$\vec{F} \neq f(\vec{r}) \vec{r},\, \text{most } \vec{r} $$
So the force vector is not proportional to the position vector, thus the problem is not a "central force" problem with the center at the origin. So it's different from problems like the Kepler problem of two objects with gravity.

 

[Muthumanimaran]

Or otherwise can I say, External torque along $${\hat{k}}$$ direction for a central force is zero. i.e, $$F_x{y}-F_y{x}=0$$, is my reasoning correct?

 

[Mark44]

the force will not be directed toward the minimum at $x=0$ & $y=0$

External torque along ${\hat{k}}$ direction

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