First slide

Cartesian plane

Question

The point (4, 1) undergoes the following three transformations successively.
(i) Reflection about the line y=x

(II) Translation through a distance 2 units along the +ve direction of x-axis

(iii) Rotation through an angle $\frac{π}{4}$ about the origin in counter clock wise direction then the final position of the point is

Moderate

A

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

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

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

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

Solution

Reflection about $y = x$ is (1, 4)
2 units along +ve direction of $x - a x i s$ is (3, 4) and using
$x' = x \cos θ - y \sin θ$ = $3 (\cos \frac{π}{4}) - 4 (\sin \frac{π}{4})$ = $\frac{3}{\sqrt{2}} - \frac{4}{\sqrt{2}}$ = $\frac{- 1}{\sqrt{2}}$
$y^{'} = x \sin θ + y \cos θ$ = $3 (\sin \frac{π}{4}) + 4 (c o s \frac{π}{4})$ = $\frac{3}{\sqrt{2}} + \frac{4}{\sqrt{2}}$ = $\frac{7}{\sqrt{2}}$

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