Question

The point (4, 1) undergoes the following three transformations successively
(i) Reflection about the line y=x
(ii)Translation through 2 units along the positive direction of X-axis
(iii)Rotation through angle π/4about the origin in the anti-clock wise sense.The coordinates of the final position of the point are

• A. $(\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}})$
• B. $(\frac{\sqrt{2}}{3},\frac{-5\sqrt{2}}{3})$
• C. (-3, 1)
• D. (-2, 2)

Answer 1

The correct option is A $(\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}})$

(i) Reflection gives the point (1, 4)
(ii) Translation along +ve X-axis by 2 units gives the point (3, 4)≡(5,θ) whereθ=tan^-1(4/3)
(iii) Rotation in the anti-clock wise sense by π/4 gives the point
$(5,\theta +\frac{\pi }{4})=\{5\cos (\theta +\frac{\pi }{4}),5\sin (\theta +\frac{\pi }{4})\}$
$=\left\{\frac{5}{\sqrt{2}}\right(\cos \theta -\sin \theta ),\frac{5}{\sqrt{2}}(\cos \theta +\sin \theta \left)\right\}$
$=\{\frac{5}{\sqrt{2}}(\frac{3}{5}-\frac{4}{5}),\frac{5}{\sqrt{2}}(\frac{3}{5}+\frac{4}{5})\}$
$=(\frac{-1}{\sqrt{2}},\frac{7}{\sqrt{2}})$