\begin{eqnarray*} & \vec{p} =( x,y,z)\\ & \vec{f} =( c,0,0) \ ,\ \vec{f'} =( -c,0,0)\\ & | | \vec{p} -\vec{f}| -| \vec{p} -\vec{f'}| | =2a\\ & | | ( x-c,y,z)| -| ( x+c,y,z)| | =2a\\ & \left| \sqrt{( x-c)^{2} +y^{2} +z^{2}} -\sqrt{( x+c)^{2} +y^{2} +z^{2}} \right| =2a\\ & \left(\sqrt{( x-c)^{2} +y^{2} +z^{2}} -\sqrt{( x+c)^{2} +y^{2} +z^{2}}\right)^{2} =4a^{2}\\ & ( x-c)^{2} +y^{2} +z^{2} -2\sqrt{\left(( x-c)^{2} +y^{2} +z^{2}\right)\left(( x+c)^{2} +y^{2} +z^{2}\right)} +( x+c)^{2} +y^{2} +z^{2} =4a^{2}\\ & 2x^{2} +2y^{2} +2z^{2} +2c^{2} -4a^{2} =2\sqrt{\left(( x-c)^{2} +\left( y^{2} +z^{2}\right)\right)\left(( x+c)^{2} +\left( y^{2} +z^{2}\right)\right)}\\ & x^{2} +y^{2} +z^{2} +c^{2} -2a^{2} =\sqrt{\left(( x-c)^{2} +\left( y^{2} +z^{2}\right)\right)\left(( x+c)^{2} +\left( y^{2} +z^{2}\right)\right)}\\ & \\ & p^{2} =x^{2} +y^{2} +z^{2}\\ & A := c^{2} -2a^{2} \\ & B := y^{2} +z^{2} \\ & p^{2} +A=\sqrt{\left(( x-c)^{2} +B\right)\left(( x+c)^{2} +B\right)}\\ & p^{4} +2Ap^{2} +A^{2} =( x-c)^{2}( x+c)^{2} +B\left(( x-c)^{2} +( x+c)^{2}\right) +B^{2}\\ & p^{4} +2Ap^{2} +A^{2} =(( x+c)( x-c))^{2} +2B\left( x^{2} +c^{2}\right) +B^{2}\\ & p^{4} +2Ap^{2} +A^{2} =\left( x^{2} -c^{2}\right)^{2} +2B\left( x^{2} +c^{2}\right) +B^{2}\\ & \\ & \left( x^{2} +y^{2} +z^{2}\right)^{2} +2\left( c^{2} -2a^{2}\right)\left( x^{2} +y^{2} +z^{2}\right) +\left( c^{2} -2a^{2}\right)^{2}\\ & = x^{4} -2c^{2} x^{2} +c^{4} +2\left( y^{2} +z^{2}\right)\left( x^{2} +c^{2}\right) +\left( y^{2} +z^{2}\right)^{2}\\ & \\ & x^{4} +y^{4} +z^{4} +2\left( x^{2} y^{2} +y^{2} z^{2} +z^{2} x^{2}\right) +2\left( c^{2} -2a^{2}\right)\left( x^{2} +y^{2} +z^{2}\right) +c^{4} -4a^{2} c^{2} +4a^{4}\\ & = x^{4} -2c^{2} x^{2} +c^{4} +2\left( x^{2} y^{2} +z^{2} x^{2}\right) +2c^{2}\left( y^{2} +z^{2}\right) +y^{4} +2y^{2} z^{2} +z^{4}\\ & \\ & x^{4} +y^{4} +z^{4} +2\left( x^{2} y^{2} +y^{2} z^{2} +z^{2} x^{2}\right) +2c^{2}\left( x^{2} +y^{2} +z^{2}\right) -4a^{2}\left( x^{2} +y^{2} +z^{2}\right) +c^{4} -4a^{2} c^{2} +4a^{4}\\ & = x^{4} +y^{4} +z^{4} +2\left( x^{2} y^{2} +y^{2} z^{2} +z^{2} x^{2}\right) +2c^{2}\left( -x^{2} +y^{2} +z^{2}\right) +c^{4}\\ & \\ & 2c^{2} x^{2} -4a^{2}\left( x^{2} +y^{2} +z^{2}\right) -4a^{2} c^{2} +4a^{4} = -2c^{2} x^{2}\\ & \\ & 4c^{2} x^{2} -4a^{2}\left( x^{2} +y^{2} +z^{2}\right) -4a^{2} c^{2} +4a^{4} =0\\ & \frac{c^{2} x^{2}}{a^{2}} -\left( x^{2} +y^{2} +z^{2}\right) -c^{2} +a^{2} =0\\ & \frac{\left( c^{2} -a^{2}\right) x^{2}}{a^{2}} -y^{2} -z^{2} =c^{2} -a^{2}\\ & \frac{x^{2}}{a^{2}} -\frac{y^{2}}{c^{2} -a^{2}} -\frac{z^{2}}{c^{2} -a^{2}} =1\\ & b^{2} := c^{2} -a^{2}\\ & \\ & \frac{x^{2}}{a^{2}} -\frac{y^{2}}{b^{2}} -\frac{z^{2}}{b^{2}} =1\\ \end{eqnarray*}