Question

"WHEN THE ORIGIN IS SHIFTED TO THE POINT (2,3) THE TRANSFORMED EQUATION OF THE CURVE IS Xsquare+3xy-2y square+17x-16xy+17y square=225 find the qriginal equatio of the curve"

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{equation}\mathrm{is}: \\ {x}^2+3xy-2y^2+17x-16xy+17y^2=225 \\ =x^2-13xy+15y^2+17x-225=0 \\ \mathrm{Here}\mathrm{we}\mathrm{want}\mathrm{to}\mathrm{shift}\mathrm{the}\mathrm{point}\left(2,3\right)\mathrm{to}\mathrm{the}\mathrm{origin}\mathrm{without}\mathrm{changing}\mathrm{the}\mathrm{direction}\mathrm{of}\mathrm{axes} \\ \mathrm{then},\mathrm{we}\mathrm{replace}x\mathrm{by}\left(x-2\right)\mathrm{and}y\mathrm{by}\left(y-3\right)\mathrm{in}\mathrm{a}\mathrm{equation}\mathrm{of}\mathrm{given}\mathrm{curve}\mathrm{then}\mathrm{the}\mathrm{required} \\ \mathrm{equation}\mathrm{is}: \\ {\left(x-2\right)}^2-3\left(x-2\right)\left(y-3\right)+15{\left(y-3\right)}^2+17\left(x-2\right)-225=0 \\ \Rightarrow x^2+4-4x-3\left(xy-3x-2y+6\right)+15\left(y^2+9-6y\right)+17x-34-225=0 \\ \Rightarrow x^2+4-4x-3xy+9x+6y-18+15y^2+135-90y+17x-259=0 \\ \Rightarrow x^2-3xy+15y^2+22x-84y-138=0 \end{gathered}$$