Question

What are represented by the equation $(r\cos \theta -a)(r-a\cos \theta )=0.$

Answer 1

$(r\cos \theta -a)(r-a\cos \theta )=0$

Using parametric form $r\cos \theta =x,r\sin \theta =y,r=\sqrt{x^2+y^2}$

$(r\cos \theta -a)(r-a\cos \theta )=0r\cos \theta -a,r-a\cos \theta =0$

$r\cos \theta -a=0r\cos \theta =a\Rightarrow x=a$

represents a vertical straight line.

$r-a\cos \theta =0r=a\cos \theta r^2=ar\cos \theta x^2+y^2=ax{x}^2+y^2-ax=0C:(\frac{a}{2},0),r=\sqrt{{\left(\frac{a}{2}\right)}^2+0-0}=\frac{a}{2}$

represents a circle with centre $C$ and radius $\frac{a}{2}$

Thus the combined equation represents a circle and a straight line.