Question

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

Answer 1

Here: $r=\sqrt{x^2+y^2}$, $x=rcos\theta$ & $y=rsin\theta$

Given equation: $r+\frac{1}{r}=3cos\theta +sin\theta$

$\Rightarrow \frac{r^2+1}{r}=3\frac{x}{r}+\frac{y}{r}$

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

$\Rightarrow x^2+y^2+1=3x+y$

$\Rightarrow x^2+y^2-3x-y+1=0$

$\because$ coefficients of x and y are same, this equation represents a $circle$.