Question

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

$4x^2+10xy+4y^2-9=0$

Short answer

The conic represented by the given equation is a hyperbola.

Step-by-step solution

Step 1. Given Information

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

$4x^2+10xy+4y^2-9=0$

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, $A=4,B=10,C=4$

Plugging the values:

$\begin{array}{rcl}{B}^2-4AC& =& {\left(10\right)}^2-4\left(4\right)\left(4\right)\\ & =& 100-64\\ & =& 36\end{array}$

Since $B^2-4AC>0$, the given conic is a hyperbola.

Step 3. Conclusion

Hence, the conic represented by the given equation is a hyperbola.