Question

What loci are represented by the equations :
$r=\alpha \sec (\theta -\alpha ).$

Answer 1

$r=a\sec (\theta -\alpha )r\cos (\theta -\alpha )=ar\cos \theta \cos \alpha +r\sin \theta \sin \alpha =a$

Using normal conventions

$x=r\cos \theta ,y=r\sin \theta \Rightarrow x\cos \alpha +y\sin \alpha =a$

So the equation represents normal form of a line whose distance from the origin is $a$