Solution:
Let the original coordinates of the point P be (x0,y0)=(4,1).
We will apply the transformations in succession and find the coordinates of P after each step.
Step (i): Origin is shifted to the point (1,6) by translation of axes.
When the origin is shifted to (h,k), the new coordinates (x′,y′) of a point (x,y) are given by the formulas:
‌x′=x−h
‌y′=y−k
In this case, (x,y)=(4,1) and (h,k)=(1,6).
So, the coordinates of P after this transformation, let's call them ( x1,y1 ), are:
‌x1=4−1=3
‌y1=1−6=−5
The point P is now at (3,−5) with respect to the new coordinate system.
Step (ii): Translation through a distance of 2 units along the positive direction of X-axis.
This implies a translation of the point itself within the current coordinate system (established after step (i)). A translation of a point (x,y) by a units along the positive X-axis results in new coordinates ( x+a,y ).
Here, the point is (x1,y1)=(3,−5), and the translation distance is a=2.
So, the coordinates of P after this transformation, let's call them (x2,y2), are:
‌x2=x1+2=3+2=5
‌y2=y1=−5
The point P is now at (5,−5).
Step (iii): Rotation of axes through an angle of 90∘ in the positive direction.
This means the coordinate axes (with respect to which P is currently at (5,−5) ) are rotated by an angle θ=90∘ counter-clockwise.
If the old coordinates of a point are (x,y) and the axes are rotated by an angle θ, the new coordinates (x′,y′) are given by:
‌x′=x‌cos‌θ+ysin‌θ
‌y′=−xsin‌θ+y‌cos‌θ
In this case, the point is (x2,y2)=(5,−5) and θ=90∘.
cos‌90∘=0
sin‌90∘=1
So, the final coordinates of P, let's call them (x3,y3), are:
‌x3=5‌cos‌90∘+(−5)sin‌90∘=5(0)+(−5)(1)=−5
‌y3=−5sin‌90∘+(−5)‌cos‌90∘=−5(1)+(−5)(0)=−5
The final position of the point P is (−5,−5).
Comparing this with the given options, the final coordinates are (−5,−5), which corresponds to Option C.
The final answer is (−5,−5).
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