Question

Identify each conic without completing the square or rotating axes.${3x}^2-xy+2y^2+3y+1=0$

Short answer

The answer is Ellipse.

Step-by-step solution

Step 1. Given information

The given equation of conic is${3x}^2-xy+2y^2+3y+1=0$

Step 2. Identify the conic without completing the square or rotating axes. 

If the equation is in the form of$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Defines,

role="math" localid="1647880942942" $$\begin{gathered} B^2-4AC=0,Parabola \\ {B}^2-4AC>0,Ellipse \\ {B}^2-4AC<0,Hyperbola \end{gathered}$$

On comparing the given equation with the above equation

$A=3;B=-1;C=2;D=0;E=3;F=1$

Step 3. Verification of conditions 

Let us verify for the conic

$\begin{array}{rcl}{B}^2-4AC& =& {\left(-1\right)}^2-4\left(3\right)\left(2\right)\\ & =& 1-24\\ & =& -23<0\end{array}$

Therefore, the given conic is Ellipse.