Question

Identify each conic without completing the square or rotating axes.$x^2-6xy+9y^2+2x-3y-2=0$

Short answer

The answer is Parabola.

Step-by-step solution

Step 1. Given information

The given equation of conic is$x^2-6xy+9y^2+2x-3y-2=0$

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="1647880928143" $$\begin{gathered} B^2-4AC=0,Parabola \\ {B}^2-4AC>0,Ellipse \\ {B}^2-4AC<0,Hyperbola \end{gathered}$$

On comparing the given equation with the above equation

$A=1;B=-6;C=9;D=2;E=-3;F=-2$

Step 3. Verification of conditions 

Let us verify for the conic

$\begin{array}{rcl}{B}^2-4AC& =& {\left(-6\right)}^2-4\left(1\right)\left(9\right)\\ & =& 36-36\\ & =& 0\end{array}$

Therefore, the given conic is Parabola.