36X^2+36y^2-24x-12y=31

Do you want to know what shape it is?   Looks like a circle to me.   

$$\\36x^2+36y^2-24x-12y=31\\ 36x^2-24x+36y^2-12y=31\\ 36(x^2-\frac{2x}{3}+y^2-\frac{1y}{3})=31\\ (x^2-\frac{2x}{3})+(y^2-\frac{1y}{3})=\frac{31}{36}\\ (x^2-\frac{2x}{3}+\frac{1}{9})+(y^2-\frac{1y}{3}+\frac{1}{36})=\frac{31}{36}+\frac{1}{9}+\frac{1}{36}\\ (x-\frac{1}{3})^2+(y-\frac{1}{6})^2=1\\$$

$$\\Circle:\qquad centre\;(\frac{1}{3},\frac{1}{6})\qquad radius=1\;unit$$

36x^2+36y^2-24x-12y=31

$\begin{array}{rcll} 36x^2+36y^2-24x-12y &=& 31 \\ (36x^2-24x)+(36y^2-12y) &=& 31 \quad & | \quad : 36 \\\\ (x^2-\frac{2}{3}x)+(y^2-\frac{1}{3}y) &=& \frac{31}{36} \\ \\ (x-\frac{2}{6})^2-\frac{4}{36}+(y-\frac{1}{6})^2-\frac{1}{36} &=& \frac{31}{36} \\ \\ (x-\frac{1}{3})^2 +(y-\frac{1}{6})^2&=& \frac{31}{36} +\frac{4}{36}+\frac{1}{36} \\ \\ (x-\frac{1}{3})^2 +(y-\frac{1}{6})^2&=& 1 \\ \\ \end{array}$

36X^2+36y^2-24x-12y=31

This is a circle with center

$\left(\dfrac{1}{3} ,\; \dfrac{1}{6}\right)$

and the radius = 1

Heureka beat me!     hi Heureka