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 aout x = 0
The final position of the given point is

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

Answer 1

The correct option is C

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

Let B, C, D, E be the positions of the given point A(4, 1) after the transformation (i), (ii), (iii) and (iv) successively.

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 $\theta +\frac{\pi }{4}$ 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, $sin\theta =\frac{4}{5},cos\theta =\frac{3}{5}$
$\Rightarrow h=\left(\frac{5}{\sqrt{2}}\right)(cos\theta -sin\theta )=-\frac{1}{\sqrt{2}}k=\left(\frac{5}{\sqrt{2}}\right)(cos\theta +sin\theta )=\frac{7}{\sqrt{2}}$
coordinates of D are $(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$ and its reflection about x = 0 in $(\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}})$