Question

What is the distance between the points $\mathrm{A}\left(\sin \theta -\cos \theta ,0\right)$ and $\mathrm{B}\left(0,\sin \theta +\cos \theta \right)$? [CBSE 2015]

Answer 1

The given points are $\mathrm{A}\left(\sin \theta -\cos \theta ,0\right)$ and $\mathrm{B}\left(0,\sin \theta +\cos \theta \right)$.

Using distance formula, we have

$$\begin{gathered} \mathrm{AB}=\sqrt{{\left[\left(\sin \theta -\cos \theta \right)-0\right]}^2+{\left[0-\left(\sin \theta +\cos \theta \right)\right]}^2} \\ =\sqrt{{\left(\sin \theta -\cos \theta \right)}^2+{\left(\sin \theta +\cos \theta \right)}^2} \\ =\sqrt{{\sin }^2\theta +{\cos }^2\theta -2\sin \theta \cos \theta +{\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta } \\ =\sqrt{2\left({\sin }^2\theta +{\cos }^2\theta \right)} \\ =\sqrt{2}\mathrm{units}\left({\sin }^2\theta +{\cos }^2\theta =1\right) \end{gathered}$$

Thus, the distance between the given points is $\sqrt{2}$ units.