16.0 Shifting of origin

  Coordinate System and Coordinates

• Let $O$ be the origin and $OX$, $OY$ be the original coordinates. Let $O'(h,k)$ be the new origin, $(h,k)$ be its coordinates and $O'X'$, $O'Y'$ be the new axes.
• Let $P(x,y)$ be any point referred to the original axes $OX$, $OY$ and coordinate of the same point $P$ with reference to the new axes $O'X'$ and $O'Y'$ be $(X, Y)$.
• Now, we have to find the relation between $x$, $h$ and $X$ and $y$, $k$ and $Y$ in order to shift the origin.
• From figure, $OL=h$, $O'L=k$, $OM=x$, $PM=y$, $O'N=X$ and $PN=Y$ and we can write $$\begin{equation} \begin{aligned} OM = OL + LM \\ OM = OL + O'N \\ x = h + X...(1) \\\end{aligned} \end{equation} $$ Similarly, $$\begin{equation} \begin{aligned} PM = PN + NM \\ PM = PN + O'L \\ y = Y + k...(2) \\\end{aligned} \end{equation} $$

• If origin $(O)$ is shifted to any point $O'(h,k)$ without rotation of axes, then new coordinates or equation of curve can be obtained by replacing $(x,y)$ with $(x+h,y+k)$ in the coordinates or equation of curve with respect to original axes.
• If the co-ordinates or equation of curve with respect to new axes are given and we have to find the co-odinates or equation of curve original axes, then replace $(x,y)$ by $(x-h,y-k)$.

Question 7. Find the equation of curve $2{x^2} + {y^2} - 3x + 5y - 8 = 0$ when the origin is transferred to the point $(-1,2)$ without changing the direction of axes.

Solution: According to the question, we have to shift the origin to the point $(h,k) \equiv ( - 1,2)$, therefore, replace $(x,y)$ with $(x+h,y+k)$ i.e., $(x-1,y+2)$ in the equation of curve with respect to original axes. Therefore, the transformed equation of curve is $$\begin{equation} \begin{aligned} 2{(x - 1)^2} + {(y + 2)^2} - 3(x - 1) + 5(y + 2) - 8 = 0 \\ 2{x^2} + {y^2} - 7x + 9y + 11 = 0 \\\end{aligned} \end{equation} $$