Question

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

I. reflection about the line $y=x$.

II. translation through a distance of $2$ units along the positive direction of $x$-axis.

III. rotation through an angle of $\frac{\mathrm{\pi }}{4}$ about the origin in the anti-clockwise direction.

The final position of the point is:

Answer 1

Find the required position of point based on given information

Given, point $(4,1)$

(i) reflection about$y=x$

Hence putting coordinates of point in given eq$y=4,x=1$

The point becomes $(1,4)$

(ii) translation through distance of $2$ units in positive $x$ - axis

Point$=(1+2,4)$

$=(3,4)$

(iii) rotation of point through and angle $\frac{\pi }{4}$​ in anticlockwise about origin $O$ After rotation Point will be $\left[r\cos (\alpha +\frac{\mathrm{\pi }}{4}),r\sin (\alpha +\frac{\mathrm{\pi }}{4})\right]$

Converting into polar form$r=\sqrt{3^2+4^2}$

$r=5$

$\therefore \tan \alpha =\frac{4}{3}$

​Hence, $\cos \alpha =\frac{3}{5}$ and $\sin \alpha =\frac{4}{5}$

Now,

$$\begin{gathered} \cos \left(\alpha +\frac{\pi }{4}\right)=\cos \alpha \cos \frac{\pi }{4}-\sin \alpha \sin \frac{\pi }{4} \\ \cos \left(\alpha +\frac{\pi }{4}\right)=\frac{3}{5}\frac{1}{\sqrt{2}}-\frac{4}{5}\frac{1}{\sqrt{2}} \\ \cos \left(\alpha +\frac{\pi }{4}\right)=\frac{-1}{5\sqrt{2}} \end{gathered}$$

and,

$$\begin{gathered} \sin \left(\alpha +\frac{\mathrm{\pi }}{4}\right)=\sin \alpha \cos \frac{\mathrm{\pi }}{4}+\cos \alpha \sin \frac{\mathrm{\pi }}{4} \\ \sin \left(\alpha +\frac{\mathrm{\pi }}{4}\right)=\frac{4}{5}\frac{1}{\sqrt{2}}+\frac{3}{5}\frac{1}{\sqrt{2}} \\ \sin \left(\alpha +\frac{\mathrm{\pi }}{4}\right)=\frac{7}{5\sqrt{2}} \end{gathered}$$

therefore, final position of the point$=\left(5\times \frac{-1}{5\sqrt{2}},5\times \frac{7}{5\sqrt{2}}\right)$

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

Hence, the final position of the point is $\left(\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}}\right)$.