Straight Line and Pair of Straight Lines 1 Question 39
39. 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
(1980, 1M)
(a) $\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}$
(b) $(-\sqrt{2}, 7 \sqrt{2)}$
(c) $-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}$
(d) $(\sqrt{2,} 7 \sqrt{2)}$
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Solution:
- 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
$$ \begin{aligned} X & =3 \cos \frac{\pi}{4}-4 \sin \frac{\pi}{4}=-\frac{1}{\sqrt{2}} \\ \text { and } \quad Y & =3 \sin \frac{\pi}{4}+4 \cos \frac{\pi}{4}=\frac{7}{\sqrt{2}} \end{aligned} $$
$\therefore$ Coordinates of $D$ are $-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}$.
