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-axis.
(iii) Rotation through an angle $\frac{\pi }{4}$ about the origin in the anticlockwise direction,
(iv) Reflection about $x=0$
The final position of the given point is

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

Answer 1

The correct option is C $(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$

Let $B,C,D,E$ be the positions of the given point $A(4,1)$ after the transformations $\left(i\right),\left(ii\right),\left(iii\right)$ and $\left(iv\right)$ successively $(Fig.15.12)$.The coordinates of $B$ are $(1,4)$ and that of $C$ are $(1+2,4+0)$ i.e. $(3,4)$. Now if $OC$ makes an angle $\theta$ with x axis, $OD$ makes and an angle $\theta \pi /4$ with x-axis, If
$(h,k)$ denote the coordinates of $D$. then $h=ODcos(\theta +{45}^{\circ }),k=\sin (\theta +{45}^{\circ })$ and $OD=OC=5\sin \theta =4/5,cos\theta =3/5$
$\Rightarrow h=\left(5/\sqrt{2}\right)(\cos \theta -sin\theta )=-1/\sqrt{2}$
$k=\left(5/\sqrt{2}\right)(\cos \theta +\sin \theta )=7/\sqrt{2}$
coordinates of $D$ are $(-1/\sqrt{2},7/\sqrt{2})$ and its reflection about $x=0$ in $(1/\sqrt{2},7/\sqrt{2})$