Question

The point $(4,1)$ undergoes the following transformations successively:

$\left(i\right)$ Reflection about the line $y=x-1$

$\left(ii\right)$ Translation by one unit along x-axis in the positive direction.

$\left(iii\right)$ Rotation through an angle $\frac{\pi }{4}$ about the origin in the anticlockwise direction.

The coordinates of $P$ in the final position are

• A. $(2,3)$
• B. $(2,3\sqrt{2})$
• C. $(0,3\sqrt{2})$
• D. None of these

Answer 1

The correct option is C $(0,3\sqrt{2})$

Let $Q(x_1,y_1)$ be the reflection of point $P(4,1)$ in the line mirror $y=x-1$. $\Rightarrow \frac{x_1-4}{1}=\frac{y_1-1}{-1}=\frac{-2\left(2\right)}{2}$

$\Rightarrow \frac{x_1-4}{1}=\frac{y_1-1}{-1}=-2$

$\Rightarrow x_1=2,y_1=3$

Thus, the coordinates of $Q$ are $(2,3)$.

Now, $Q$ is translated parallel to x-axis in the positive direction by $1$ unit.

So, the coordinates of $R$ are $(1+2,3)$ or $(3,3)$.

Suppose $OR$ makes an angle $\theta$ with $OX$. Then,

$r=\sqrt{3^2+3^2}=3\sqrt{2}$

$\tan \theta =\frac{3}{3}=1\Rightarrow \theta =\frac{\pi }{4}$

It is given that $OR$ is rotated through $\frac{\pi }{4}$ in anticlockwise sense to coincide with $OS$. Therefore, $OS$ makes a right angle with $OX$ and $OS=OR=3\sqrt{2}$

Thus, the coordinates of $S$ are $(OS\cos \frac{\pi }{2},OS\sin \frac{\pi }{2})$ i.e. $(0,3\sqrt{2}).$