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| Translation of Axes |
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| An equation corresponding to a set of points with reference to a system of coordinate axes may be simplified by taking the set of points in some other suitable coordinate system, such that all geometrical properties remain unchanged. One such transformation is that in which the new axes are transformed parallel to the original axes and origin is shifted to a new point. A transformation of this kind is called a translation of axes. |
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| The coordinates of each point of the plane are changed under a translation of axes. By knowing the relationship between the old coordinates and the new coordinates of points, we can study the analytical problem in terms of new system of coordinate axes. |
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| To see how the coordinates of a point of the plane can change under a translation of axes, let us take a point P(x,y) referred to the axes OX and OY. Let O'X' and O'Y' be new axes parallel to OX and OY, respectively, where O' is the new origin. Let (h,k) be the coordinates of O' referred to the old axes, i.e., OL = h and LO' = k. Also, OM = x and MP = y . |
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| Let O'M' = x' and M'P = y' be respectively, the abscissa and ordinates of point P referred to the new axes O'X' and O'Y'. |
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| From the figure, we have |
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| OM = OL + LM = OL + O'M', i.e., x = h + x' |
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| MP = MM' + M'P = LO' + M'P, i.e., y = k+ y' |
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| Hence x = x' + h, y = y' + k. |
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| These formulae give the relations between the old and new coordinates. Therefore, if the equation of the set of points P (locus of P) with respect to OX and OY be f (x,y) = 0, the equation to the same set of points when the origin is translated to O', becomes f (x' + h, y' + k) = 0, where x', y' are coordinates with reference to the new axes O'X' and O'Y'. |
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