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$x^{2}y+xy^{2}+54=0$
differentiating with respect to x,
$2xy+x^{2}(\frac{dy}{dx})+y^{2}+2xy(\frac{dy}{dx})=0$
When $\frac{dy}{dx}=-1$
$-x^{2}+2xy+y^{2}-2xy=0$
$y^{2}=x^{2}$
$y=\pm{x}$

When $y=x$,
$y^{3}+y^{3}+54=0$
$y^{3}=-27$
$y=-3$
$x=-3$

When $y=-x$
$-y^{3}+y^{3}+54=0$
There is no solutions.

Therefore, required coordinates are $(-3,-3)$

This question tests students on their understanding of basic differentiations. First pitfall was when students forget to introduce $\pm{ }$. And to SHOW there is only one such point baffled many as they didn’t know how to go about. But the most direct way is to solve for them and find there is only one. Lastly, read the question carefully and you will see they require the answers to be COORDINATES.