Question

Find the distance between the points $(a\sin \theta ,a\cos \theta )$ and $(a\cos \theta ,-a\sin \theta )$

Answer 1

Distance between the points $(a\sin \theta ,a\cos \theta )$ and $(a\cos \theta ,-a\sin \theta )$
$d=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$
$=\sqrt{(a\sin \theta -a\cos \theta {)}^2+(a\cos \theta +a\sin \theta {)}^2}$
$=\sqrt{a^2(\sin \theta -\cos \theta {)}^2+a^2(\cos \theta +\sin \theta {)}^2}$
$=a\sqrt{{\sin }^2\theta +{\cos }^2\theta -2\sin \theta \cos \theta +{\cos }^2\theta +{\sin }^2\theta +2\sin \theta \cos \theta }$
$=a\sqrt{({\sin }^2\theta +{\cos }^2\theta )+({\cos }^2\theta +{\sin }^2\theta )}$
$=a\sqrt{1+1}$
$=a\sqrt{2}$