Question

What are represented by the equation $\frac{1}{r}=1+\cos \theta +\sqrt{3}\sin \theta$

Answer 1

$\frac{1}{r}=1+\cos \theta +\sqrt{3}\sin \theta 1=r+r\cos \theta +r\sqrt{3}\sin \theta$

Now we know $r\cos \theta =x,r\sin \theta =y,r=\sqrt{x^2+y^2}$

$\Rightarrow 1=\sqrt{x^2+y^2}+x+\sqrt{3}y\Rightarrow \sqrt{x^2+y^2}=1-x-\sqrt{3}y$

Squaring both sides

$x^2+y^2=1+x^2+3y^2-2x+2\sqrt{3}xy-2\sqrt{3}y2y^2+2\sqrt{3}xy-2x-2\sqrt{3}y+1=0$

Comparing the equation with general equation of second degree

$\Rightarrow a=0,b=2,h=\sqrt{3},g=-1,f=-\sqrt{3},c=1\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2\Rightarrow \Delta =(0\times 2\times 1)+(2\times -\sqrt{3}\times -1\times \sqrt{3})-(0\times 3)-(2\times 1)-(1\times 3)\Rightarrow \Delta =0+6-2-3\Rightarrow \Delta =1\Rightarrow \Delta >0$

So the equation represents a hyperbola.