In many cases, to solve a problem it may be useful to shift the axes and the origin to a new point to suit your calculations. This approach can drastically reduce the complexity of a problem.
Let's say we have been given the points $A\ (8,\ 13)$ and $B\ (18,\ -11)$ and are required to find the distance between the two points.
We can of course find it the traditional way by calculating:
$dist = \sqrt{(8 - 18)^2 + (13 + 11)^2}$
$= \sqrt{(-10)^2 + (24)^2}$
$ = \sqrt{676} = 26$
Now, if we take the two axes and move them anywhere, the absolute coordinates of these points will change, but the distance between them will remain same since the relative positions of the two points (that is, the position of one point with respect to the other) will not change.
Will this help our calculation? It can if we choose the new origin smartly.
Let us choose point $A$ as the new origin, that way one of our points will become $(0, 0)$.
So we take the new origin $O'$ and move it to the point $(8, 13)$.
The new axes formed by this move is shown in the following figure as $X'AY'$
In the new coordinate system $X'Y'$,
$A = (8 - 8, 13 -13) = (0, 0)$, and
$B = (18 - 8, -11 -13) = (10, -24)$
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The distance can be calculated using the same formula as $\sqrt{10^2 + (-24)^2} = 26$
Now try the following widget. Try moving the axes $X'Y'$ using the blue dot or the line using the two red dots. See what all changes or does not change when you move just the axes or the line itself.
(You may want to maximise this widget to full screen mode to get a clearer view)
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Since moving the axes around does not change the relative position of the two points $A$ and $B$, the length of $\overline{AB}$ will not change. Similarly, since the new axes will be parallel to the original axes, the slope of $AB$ will also not change.
However, observe that the equation of the line $AB$ changes as you move the origin around, because the coordinates of every point on the line will change w.r.t the new origin.
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Now let us try the following question:
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This was a simple example to illustrate the usefulness of the origin shifting. We will see many more uses of this technique and much larger benefits of axes translation as we practice more problems.
Now see if you can attempt the following problem:
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II. Equations Wtih Axis Translation
We have seen in the previous section how coordinates of any given point changes with axes translation. In this section we will see how an equation changes when the axes of the coordinate system are translated.
We will take a linear equation to explain this, however, this approach will work for any generic equation as well.
Let us take a linear equation $ax + by + c = 0$ on a Cartesian plane. This equation represents an infinite number of points, each identified by a pair of $(x,\ y)$ values.
Now let us apply a translation of amount $t_x$ and $t_y$ to the $X$ and $Y$ axes respectively.
As we saw in the previous section, this is equivalent of shifting all the points by amounts $-t_x$ and $ -t_y$ in the $X$ and $Y$ directions respectively.
If we represent the new coordinates of the points on the given line as $x'\ y'$, then, we get:
$x' = x - t_x$ and $y' = y - t_y$
$\therefore x = x' + t_x$ and $y = y' + t_y$
Substituting these values in our original equation $ax + by + c = 0$ we get:
$a(x' + t_x) + b(y' + t_y) + c = 0$
$\Rightarrow ax' + at_x + by' + bt_y + c = 0$
Since, $a$, $b$, $t_x$ and $t_y$ are constants, merging them together, we get:
$ax' + by' + (at_x + bt_y + c) = 0$
If we calculate the slope of the two given lines using the method described , we will see that both lines have a slope of $-\dfrac{a}{b}$
It is very intuitive and easy to visualise that if we perform only a translation of the axes, without rotating them, then the slope of the new equation will remain the same as that of the original line.
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Now, let us try the following problem:
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