Shift of Origin

5.1.2 Shift of Origin

Let $O'(h,k)$ be a chosen point in the $XY$-plane, and imagine sliding the whole coordinate system so that the origin moves to $O'$, while the new axes $O'X'$ and $O'Y'$ stay parallel to the original axes $OX$, $OY$. A point $P$ in the plane then has two coordinate pairs: $(x,y)$ measured from the old axes, and $(x',y')$ measured from the new axes. We want the relation between them.

Derivation. Drop $PL \perp OX$, meeting $O'X'$ at $L'$; drop $PM \perp OY$, meeting $O'Y'$ at $M'$. Let $O'Y'$ meet $OX$ at $N$, and let $O'X'$ meet $OY$ at $T$. By construction, $ON = h$, $OT = k$, $OL = x$, $OM = y$, $O'L' = x'$, $O'M' = y'$. Now

So the shift-of-origin formulas are

(equivalently, using capital letters $(X,Y)$ or $(u,v)$ for the new coordinates instead of $(x',y')$ is just a naming choice — the formulas are the same).

Worked Example 1. Origin shifted to $O'(3,2)$; find the new coordinates of $A(4,6)$ and $B(2,-5)$. Here $x = x'+3$, $y=y'+2$. For $A$: $4 = x'+3 \Rightarrow x'=1$; $6=y'+2 \Rightarrow y'=4$. New coordinates of $A$ are $(1,4)$. For $B$: $2=x'+3\Rightarrow x'=-1$; $-5=y'+2\Rightarrow y'=-7$. New coordinates of $B$ are $(-1,-7)$.

Worked Example 2. Origin shifted to $(-2,1)$, axes parallel to the original; new coordinates of $A$ are $(7,-4)$ — find the old coordinates. Here $h=-2,k=1$, so $x=X-2$, $y=Y+1$. With $(X,Y)=(7,-4)$: $x=7-2=5$, $y=-4+1=-3$. Old coordinates of $A$ are $(5,-3)$.

Worked Example 3. Find the new equation of the locus $x^2 - xy - 2y^2 - x + 4y + 2 = 0$ when the origin is shifted to $(2,3)$. Here $h=2,k=3$, so $x = X+2$, $y=Y+3$. Substituting throughout, …