Question

The point $A(4,1)$ undergoes following transformations successively :
(i) reflection about line $y=x$
(ii) translation through a distance of $3$ units in the positive direction of $x$ -axis
(iii) rotation through an angle ${105}^{\circ }$ in anti-clockwise direction about origin $O$.
Then the final position of point $A$ is

• A. $(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$
• B. $(-2,7\sqrt{2})$
• C. $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$
• D. $(-2\sqrt{6},2\sqrt{2})$

Answer 1

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

(i) point A(4,1) undergoes reflection about y = x

$\therefore$ coordinate A becomes (1, 4)

Now,

(ii) it undergoes translation through distance of

3 unit in positive direction of x - axis

so new coordinate is (4, 4)

(iii) Rotated through an angle of ${105}^{\circ }$ in axis

clock - wise direction.

$\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]\left[\begin{array}{c}4\\ 4\end{array}\right]$

$=\left[\begin{array}{c}4(cos\theta -sin\theta )\\ 4(sin\theta +cos\theta )\end{array}\right]$

$=\left[\begin{array}{c}4(cos105-sin105)\\ 4(sin105+cos105)\end{array}\right]$

$=\left[\begin{array}{c}-2\sqrt{6}\\ 4\times \frac{1}{\sqrt{2}}\end{array}\right]$

$=\left[\begin{array}{c}-2\sqrt{6}\\ 2\sqrt{2}\end{array}\right]$

final position $A(-2\sqrt{6},2\sqrt{2})$