Question

The point (4, 1) undergoes the following three transformations successively I. Reflection about the line y = x. II. Transformation 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 counter clockwise direction. Then, the final position of the point is given by the coordinates

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

Answer 1

Answer: (C)

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

Answer 2

Let B, C, D be the position of the point A (4,1) after the three operations I, II and III, respectively. Then, B is (1, 4), C(1 + 2,4) i.e. (3, 4). The point D is obtained from C by rotating the coordinate axes through an angle $\pi/4$ in anti-clockwise direction. Therefore, the coordinates of D are given by $X =3cos \frac{\pi}{2}-4 sin \frac{\pi}{4}=-\frac{-1}{\sqrt{2}}$ and $Y=3 sin \frac{\pi}{4}+4 cos\frac{\pi}{4}=\frac{7}{\sqrt{2}}$ $\therefore$ Coordinates of D are $\Bigg(-\frac{1}{\sqrt{2}},\frac{7}{\sqrt{2}}\Bigg).$