${\displaystyle {\displaystyle {\frac {a^{2}x^{2}}{x^{2}+y^{2}+z^{2}-a^{2}w^{2}}}+{\frac {b^{2}y^{2}}{x^{2}+y^{2}+z^{2}-b^{2}w^{2}}}+{\frac {c^{2}z^{2}}{x^{2}+y^{2}+z^{2}-c^{2}w^{2}}}=0}}$ With ${\displaystyle a=1,b=0.5,c=1.5,w=1}$.

Multiplying out the denominator and removing common terms give the quartic ${\displaystyle (a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2})(x^{2}+y^{2}+z^{2})-(a^{2}x^{2}b^{2}w^{2}+a^{2}x^{2}c^{2}w^{2}+a^{2}w^{2}b^{2}y^{2}+b^{2}y^{2}c^{2}w^{2}+a^{2}w^{2}c^{2}z^{2}+b^{2}w^{2}c^{2}z^{2})+a^{2}b^{2}c^{2}w^{4}=0}$