Question

Identify each conic without completing the square or rotating axes.$2x^2+5xy+3y^2+3x-7=0$

Short answer

The required answer is Hyperbola.

Step-by-step solution

Step 1. Given information

The given equation of conic is$2x^2+5xy+3y^2+3x-7=0$

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="1647880952759" $$\begin{gathered} B^2-4AC=0,Parabola \\ {B}^2-4AC>0,Ellipse \\ {B}^2-4AC<0,Hyperbola \end{gathered}$$

On comparing the given equation with the above equation

$A=2;B=5;C=3;D=3;E=0;F=-7$

Step 3. Verification of conditions 

Let us verify for the conic

$\begin{array}{rcl}{B}^2-4AC& =& {\left(5\right)}^2-4\left(2\right)\left(3\right)\\ & =& 25-24\\ & =& 1>0\end{array}$

Therefore, the given conic is Hyperbola.