Question

The maximum distance between the points $(a\cos \alpha ,a\sin \alpha )$ and $(a\cos \beta ,a\sin \beta )$ is

• A. $|a|\text{units}$
• B. $4|a|\text{units}$
• C. $3|a|\text{units}$
• D. $2|a|\text{units}$

Answer 1

The correct option is D $2|a|\text{units}$

Distance between the given two points
$=\sqrt{(a\cos \alpha -a\cos \beta {)}^2+(a\sin \alpha -a\sin \beta {)}^2}=\sqrt{a^2\left[\right(\cos \alpha -\cos \beta {)}^2+(\sin \alpha -\sin \beta {)}^2]}=|a|\sqrt{({\cos }^2\alpha +{\sin }^2\alpha )+({\cos }^2\beta +{\sin }^2\beta )-2(\cos \alpha \cos \beta +\sin \alpha \sin \beta )}=|a|\sqrt{2-2\cos (\alpha -\beta )}=|a|\sqrt{2[1-\cos (\alpha -\beta \left)\right]}=|a|\sqrt{4{\sin }^2\left(\frac{\alpha -\beta }{2}\right)}=2\left|a\sin \left(\frac{\alpha -\beta }{2}\right)\right|$

We know that,
$\sin \left(\frac{\alpha -\beta }{2}\right)\in [-1,1]$
Hence, the maximum distance $=2|a|\text{units}$