Question

If the distance from $(1,0)$ to two distinct points $A(\sin \theta ,\cos \theta )$ and $B({\sin }^2\theta ,-\cos \theta )$ is equal$(\ne 0)$ then the distance $AB$ (in units) is

Answer 1

Since distance are equal :
$\Rightarrow (\sin \theta -1{)}^2+{\cos }^2\theta =({\sin }^2\theta -1{)}^2+{\cos }^2\theta$
$\Rightarrow {\sin }^2\theta -1=\pm (\sin \theta -1)$
$\Rightarrow {\sin }^2\theta =\sin \theta$ or ${\sin }^2\theta +\sin \theta =2$
$\therefore \sin \theta =0$ or $\sin \theta =1$
$\sin \theta =1$ is not possible as points become coincident in this case
$\therefore \sin \theta =0\Rightarrow A(0,1)$ and $B(0,-1)\text{or}B(0,1)$ and $A(0,-1)$
$\Rightarrow AB=2$ units