The point P(4,1) undergoes the following tran | vedclass

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(C) Let the initial coordinates be $(x, y) = (4, 1)$.$(i)$ After shifting the origin to $(1, 6)$,the new coordinates $(x', y')$ are given by $x' = x -

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$(-5,-5)$

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(C) Let the initial coordinates be $(x, y) = (4, 1)$.$(i)$ After shifting the origin to $(1, 6)$,the new coordinates $(x', y')$ are given by $x' = x - 1 = 4 - 1 = 3$ and $y' = y - 6 = 1 - 6 = -5$. So,$P' = (3, -5)$.(ii) After translation by $2$ units along the positive $X$-axis,the new coordinates $(x'', y'')$ are $x'' = x' + 2 = 3 + 2 = 5$ and $y'' = y' = -5$. So,$P'' = (5, -5)$.(iii) After rotation of axes by $90^{\circ}$ in the positive direction,the new coordinates $(X, Y)$ are given by $X = x'' \cos(90^{\circ}) + y'' \sin(90^{\circ}) = 5(0) + (-5)(1) = -5$ and $Y = -x'' \sin(90^{\circ}) + y'' \cos(90^{\circ}) = -5(1) + (-5)(0) = -5$.Thus,the final coordinates are $(-5, -5)$.

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