Show That the Curves $x ^ { 3 } - 3 x y ^ { 2 } + y = 0$

question 234

Essay

Show that the curves $x ^ { 3 } - 3 x y ^ { 2 } + y = 0$ and $3 x ^ { 2 } y - y ^ { 3 } - x = 4$ are orthogonal.

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Definitions:

Valence Shells

The outermost shell of an atom that contains the electrons involved in forming bonds.

Noble Gas Elements

Elements in group 18 of the periodic table, known for their lack of reactivity due to having full valence electron shells.

Electrons

Negatively charged subatomic particles found in all atoms, orbiting the nucleus and involved in chemical bonds and electricity.

Octet Rule

A chemical rule of thumb that atoms of main-group elements tend to combine in such a way that each atom has eight electrons in its valence shell, giving it the same electron configuration as a noble gas.