Para encontrarmos a curva de encontro, primeiramente vamos esboçar as duas funções no R3:

\(\begin{align} & {{\left( x\text{ }-\text{ }2 \right)}^{2}}\text{ }+\text{ }{{\left( y\text{ }-\text{ }3 \right)}^{2}}\text{ }+\text{ }{{\left( z\text{ }+\text{ }5 \right)}^{2}}\text{ }=\text{ }32 \\ & \text{ }x\text{ }+\text{ }y\text{ }+\text{ }z\text{ }=\text{ }0 \\ & (1,1,1) \\ & \\ & C=1{{\left( x\text{ }-\text{ }2 \right)}^{2}}\text{ }+1\text{ }{{\left( y\text{ }-\text{ }3 \right)}^{2}}\text{ }+\text{ 1}{{\left( z\text{ }+\text{ }5 \right)}^{2}}\text{ }-\text{ }32 \\ & C={{\left( x\text{ }-\text{ }2 \right)}^{2}}\text{ }+\text{ }{{\left( y\text{ }-\text{ }3 \right)}^{2}}\text{ }+{{\left( z\text{ }+\text{ }5 \right)}^{2}}\text{ }-\text{ }32 \\ \end{align}\ \)

Portanto, a curva de interseção será:

\(\begin{align} & C={{\left( x\text{ }-\text{ }2 \right)}^{2}}\text{ }+\text{ }{{\left( y\text{ }-\text{ }3 \right)}^{2}}\text{ }+{{\left( z\text{ }+\text{ }5 \right)}^{2}}\text{ }-\text{ }32 \\ \end{align}\ \)