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The point $A\left( 4,1 \right)$ undergoes following transformations successively
(i) reflection about line $y=x$.
(ii) translation through a distance of 2 units in the positive direction of x axis.
(iii) rotation through an angle $\dfrac{\pi }{4}$ in anti-clock wise direction about origin O.
Then the final position of point A is
(a) $\left( \dfrac{1}{\sqrt{2}},\dfrac{7}{\sqrt{2}} \right)$
(b) $\left( -2,7\sqrt{2} \right)$
(c) $\left( \dfrac{7}{\sqrt{2}},\dfrac{1}{\sqrt{2}} \right)$
(d) None of these

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Last updated date: 13th Jun 2024
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Answer
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Hint:
 We start solving the problem by finding the position of the given point after making a reflection against the line $y=x$. We then make use of the fact that if axes are translated h units along with x-axis and k units along the y-axis, then the new position of co-ordinates will be $x'=x+h$, $y'=y+k$ to get the new position of the point. We then make use of the fact that if axes are rotated through an angle $\theta $ anti-clock wise, then the new position of co-ordinates will be $x'=x\cos \theta +y\sin \theta $, $y'=-x\sin \theta +y\cos \theta $ to get the final position of the point.

Complete step by step answer:
According to the problem, we are asked to find the final position the point $A\left( 4,1 \right)$ after undergoing the following transformations successively:
(i) reflection about line $y=x$.
(ii) translation through a distance of 2 units in the positive direction of x-axis.
(iii) rotation through an angle $\dfrac{\pi }{4}$ in anti-clockwise direction about origin O.

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Let us assume B be the position of the point $A\left( 4,1 \right)$ after undergoing reflection about line $y=x$ which will be just the interchange of co-ordinates in the given point. So, the new position is $B\left( 1,4 \right)$.
Let us assume C be the position of the point $B\left( 1,4 \right)$ after undergoing translation through a distance of 2 units in the positive direction of x axis.
We know that if axes are translated h units along x-axis and k units along y-axis, then the new position of co-ordinates will be $x'=x+h$, $y'=y+k$.
So, the new position of point B is $C\left( 1+2,4 \right)=C\left( 3,4 \right)$.
Let us assume D be the position of the point $C\left( 3,4 \right)$ after undergoing rotation through an angle $\dfrac{\pi }{4}$ in anti-clock wise direction about origin O.
We know that if axes are rotated through an angle $\theta $ anti-clock wise, then the new position of co-ordinates will be $x'=x\cos \theta +y\sin \theta $, $y'=-x\sin \theta +y\cos \theta $.
So, the new position of point C is $D\left( \left( 3\cos \dfrac{\pi }{4}+4\sin \dfrac{\pi }{4} \right),\left( -3\sin \dfrac{\pi }{4}+4\cos \dfrac{\pi }{4} \right) \right)=D\left( \dfrac{7}{\sqrt{2}},\dfrac{1}{\sqrt{2}} \right)$.
$\therefore $ The correct option for the given problem is (c).

Note:
We should not confuse the old and new positions of the given position while translating and rotation of the axes of the problem. We should not confuse rotation with translation or vice-versa, as this is the most common mistake done by students. Similarly, we can expect problems to find the new position D if the axes are rotated by $\dfrac{\pi }{2}$ clockwise later.