Question

The point (4, 1) undergoes the following transformation successively.
(i)reflection about the line y=x
(ii)translation through a distance 2 units along the positive direction of x-axes.
(iii)rotation through an angle $\pi /4$ about the origin in the anticlockwise direction.
(iv) reflection about x=0
The final position of the given point is

• A. $(1\sqrt{2},7/2)$
• B. $(1/2,7\sqrt{2})$
• C. $(1\sqrt{2},7/\sqrt{2})$
• D. $(1/2,7/2)$

Answer 1

The correct option is A $(1\sqrt{2},7/2)$

$\text{Solve: Given,}p(4,1)$

$\text{step I: Reflection of}P(1,4)\text{about}y=x$

$\Rightarrow p^{\prime }(1,4)$

Step II: translation by 2 unit in positive

direction of $x$ -axis

hew coordinate $(1+2,4)$

i.e. (3,4)

Step III: Rotation through angle $\pi /4$ about

origin in anticlockwise direction.

$\Rightarrow \left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$

Given, $\theta =\frac{\pi }{4}$ and $x=3,y=4$

$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \frac{\pi }{4}& \sin \frac{\pi }{4}\\ -\sin \frac{\pi }{4}& \cos \frac{\pi }{4}\end{array}\right]\left[\begin{array}{c}3\\ 4\end{array}\right]$

$x^{\prime }=3\cos \frac{\pi }{4}+4\sin \frac{\pi }{4}=\frac{7}{\sqrt{2}}$

$y^{\prime }=-3\sin \frac{\pi }{4}+4\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}$

$(\frac{7}{\sqrt{2}},\frac{1}{\sqrt{2}})$

Step: I V Ref. about $x=0$

i.e. Reflection about $y$ axis

$\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}]=\left[\begin{array}{c}\cos \frac{x}{4}-\sin \frac{x}{4}\\ -\sin \frac{\pi }{4}\cos \frac{\pi }{4}\end{array}\right]\left[\begin{array}{c}3\\ 4\end{array}\right]$

$x^{\prime }=3\cos \frac{x}{4}-4\sin \frac{\pi }{4}=-\frac{1}{\sqrt{2}}$

$y^{\prime }=-3\sin \frac{\pi }{4}+4\cos \frac{\pi }{4}=\frac{7}{\sqrt{2}}$

$(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$

In step IV: Reflection about $x=0$

means about y axis so, sign

of $x$ coordinate change.

final coordinate is $(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$