Question

Draw the loci
(1) $r=a$; (2) $r=a\sin \theta$; (3) $r=\alpha \cos \theta$; (4) $r=\alpha \sec \theta$; (5) $r=a\cos (\theta -\alpha )$; (6) $r=a\sec (\theta -\alpha )$.

Answer 1

(1)

$r=a$

$x=r\cos \theta$

$y=r\sin \theta$

$x^2+y^2=a^2$

(2)

$r=a\sin \theta$

$\Rightarrow x=r\cos \theta$

$\Rightarrow y=r\sin \theta$

$r^2=ay$

$x^2+y^2=ay$

$\Rightarrow x^2+{(y-\frac{a}{2})}^2={\left(\frac{a}{2}\right)}^2$

(3)

$r=a\cos \theta$

$x=r\cos \theta$

$y=r\sin \theta$

$x^2+y^2=r^2$

$r^2=ax$

$x^2+y^2=ax$

${(x-\frac{a}{2})}^2+y^2={\left(\frac{a}{2}\right)}^2$

(4)

$r=\frac{a}{cos\theta }$

$a=r\cos \theta$

$x=a$

(5)

$r=a\cos (\theta -\alpha )$

$x=r\cos \theta$

$y=r\sin \theta$

$r=a\cos \theta \cos \alpha +a\sin \theta \sin \alpha$

$r=\frac{ax}{r}\cos \alpha +\frac{ay}{r}\sin \alpha$

$r^2=ax\cos \alpha +ay\sin \alpha$

${(x-\frac{a}{2}\cos \alpha )}^2+{(y-\frac{a}{2}\sin \alpha )}^2={\left(\frac{a}{2}\right)}^2$

(6)

$r=\frac{a}{\cos (\theta -\alpha )}$

$x=r\cos \theta$

$y=r\sin \theta$

$r=\frac{a}{\cos \theta \cos \alpha +\sin \theta \sin \alpha }$

$x\cos \alpha +y\sin \alpha =a$