Question

Find the distances between the following pair of points. $(acos\alpha ,asin\alpha )$ and $(\alpha cos\beta ,\alpha sin\beta )$

Answer 1

Distance $d$ between two points $P(a,b)$ and $Q(c,d)$ is given by

$d=\sqrt{(a-c{)}^2+(b-d{)}^2}$

Given points
$(acos\alpha ,asin\alpha )$ &$(acos\beta ,asin\beta )$
$d=\sqrt{(acos\alpha -acos\beta {)}^2+(asin\alpha -asin\beta {)}^2}$

$\Rightarrow d=a\sqrt{co{s}^2\alpha +si{n}^2\alpha +co{s}^2\beta +si{n}^2\beta -2(cos\alpha cos\beta +sin\alpha sin\beta )}$

$\Rightarrow d=a\sqrt{2-2cos(\alpha -\beta )}$
$\Rightarrow d=a\sqrt{2(1-cos(\alpha -\beta \left)\right)}$
$\Rightarrow d=a\sqrt{2\times 2si{n}^2\left(\frac{\alpha -\beta }{2}\right)}$
$\Rightarrow d=2asin\frac{\alpha -\beta }{2}$
formula used
$1-cos\varphi =2si{n}^2\frac{\varphi }{2}$
$cos(\varphi -\Theta )=cos\varphi cos\Theta -sin\varphi sin\Theta$