The transformed equation of $x^2–y^2+2x+4y=0$  when the origin is shifted to the point (–1, 2) is

$$\begin{gathered} \mathrm{Given} \mathrm{origin} \mathrm{shifted} \mathrm{to} \mathrm{the} \mathrm{point} (\mathrm{h},\mathrm{k})=(-1,2) \\ \mathrm{Given} \mathrm{original} \mathrm{equation} {\mathrm{x}}^2-{\mathrm{y}}^2+2\mathrm{x}+4\mathrm{y}=0 \\ \mathrm{Now} \mathrm{transformed} \mathrm{equation} \mathrm{is} \\ {(X-1)}^2-{(Y+2)}^2+2(X-1)+4(Y+2)=0 \\ \Rightarrow (X^2+1-2X)-(Y^2+4+4Y)+2X-2+4Y+8=0 \\ \Rightarrow X^2+1-2X+Y^2-4-4Y+2X-2+4Y+8=0 \\ \Rightarrow X^2-Y^2+3=0 \end{gathered}$$