Question

In Problems 19 –23, identify each conic without completing the squares and without applying a rotation of axes.

$x^2-2xy+3y^2+2x+4y-1=0$

Short answer

The conic represented by the given equation is an ellipse.

Step-by-step solution

Step 1. Given Information

Given to identify the conic without completing the squares and without applying a rotation of axes.

$x^2-2xy+3y^2+2x+4y-1=0$

Step 2. Calculation

An equation of the form $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$defines:

• a parabola if $B^2-4AC=0$
• an ellipse/ a circle if $B^2-4AC<0$
• a hyperbola if $B^2-4AC>0$

In the given equation, $A=1,B=-2,C=3$

Plugging the values,

$\begin{array}{rcl}{B}^2-4AC& =& {\left(-2\right)}^2-4\left(1\right)\left(3\right)\\ & =& 4-12\\ & =& -8\end{array}$

Since $B^2-4AC<0$, the given conic is an ellipse or a circle.

Since$A\ne C$, the conic is an ellipse.

Step 3. Conclusion

Hence, the conic represented by the given equation is an ellipse.