Question

If a point $P(4,3)$ is rotated through an angle ${45}^o$ in anticlockwise direction about origin, then co-ordinates of $P$ in new position are :

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

Answer 1

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

$\begin{array}{c}{m}_2=\tan \left(\theta +{45}^0\right)=\frac{\tan \theta +1}{1-\tan \theta }=\frac{\frac{3}{4}+1}{1-\frac{3}{4}}\\ =\frac{\frac{7}{4}}{\frac{1}{4}}=7\\ \alpha =\theta +{45}^0\\ \tan \alpha =7\\ \frac{x-0}{\cos \alpha }=\frac{y-0}{\sin \alpha }=5\\ x=\frac{5}{\sqrt{501}}y=\frac{5\times 7}{\sqrt{501}}=\frac{35}{\sqrt{0}}\\ x=\frac{1}{\sqrt{2}}y=\frac{7}{\sqrt{2}}\end{array}$

Option $A$ is correct answer.