The transformed equation of ${\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2-6\mathrm{x}-8\mathrm{y}+2\mathrm{z}+24=0$ when the axes are translated to the point (3, 4, -1) is

$$\begin{gathered} {\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2-6\mathrm{x}-8\mathrm{y}+2\mathrm{z}+24=0 \\ \left(x^2-6x+9\right)+\left(y^2-8y+16\right)+\left({\mathrm{z}}^2+2\mathrm{z}+1\right)=2 \\ {\left(x-3\right)}^2+{\left(y-4\right)}^2+{\left(z+1\right)}^2=2 \\ \mathrm{Axes} \mathrm{are} \mathrm{translated} \mathrm{to} \mathrm{the} \mathrm{point}\left(3,4,-1\right) \\ \Rightarrow \mathrm{x}=\mathrm{X}+3,\mathrm{y}=\mathrm{Y}+4,\mathrm{z}=\mathrm{Z}-1 \\ \therefore \mathrm{The} \mathrm{transformed} \mathrm{equation} \mathrm{is} {\mathrm{X}}^2+{\mathrm{Y}}^2+{\mathrm{Z}}^2=2 \end{gathered}$$