Question

Let $A=(\sin \theta +1,\cos \theta )$ and $B=(1-\cos \theta ,-\sin \theta )$. Then the maximum value of $AB$ is

• A. $2$
• B. $\sqrt{2}$
• C. $3$
• D. $\sqrt{3}$

Answer 1

The correct option is A $2$

$AB=\sqrt{(\sin \theta +\cos \theta {)}^2+(\sin \theta +\cos \theta {)}^2}=\sqrt{2}\left(\sqrt{1+\sin 2\theta }\right)$

$\therefore A{B}_{max}=2$, possible when $2\theta =\frac{\pi }{2}\Rightarrow \theta =\frac{\pi }{4}$