$$\eqalign{
& Lay\,\,M\left( {x;y} \right) \in \left( C \right),\,\,M'\left( {x';y'} \right) = {T_{\overrightarrow v }}\left( M \right) \cr
& BTTD:\,\,\left\{ \matrix{
x' = x - 2 \hfill \cr
y' = y + 5 \hfill \cr} \right. \Rightarrow \left\{ \matrix{
x = x' + 2 \hfill \cr
y = y' - 5 \hfill \cr} \right. \cr
& \Rightarrow M\left( {x' + 2;y' - 5} \right) \in \left( C \right) \cr
& \Rightarrow {\left( {x' + 2} \right)^2} + {\left( {y' - 5} \right)^2} - 2\left( {x' + 2} \right) + 4\left( {y' - 5} \right) - 4 = 0 \cr
& \Leftrightarrow x{'^2} + 4x' + 4 + y{'^2} - 10y' + 25 - 2x' - 4 + 4y' - 20 - 4 = 0 \cr
& \Leftrightarrow x{'^2} + y{'^2} + 2x' - 6y' + 1 = 0 \cr
& {T_{\overrightarrow v }}\left( C \right) = \left( {C'} \right) \Rightarrow M' \in \left( {C'} \right) \cr
& \Rightarrow \left( {C'} \right):\,\,{x^2} + {y^2} + 2x - 6y + 1 = 0 \cr} $$
