Chapter A5.2, Problem 11E

To determine

whether the given equations represent parabolas, ellipses, or hyperbolas.

Answer to Problem 11E

Since $D>0$ , the given curve represents a hyperbola.

Explanation of Solution

Given:

  $3x^2-5xy+2y^2-7x-14y=-1$

Formula Used:

The formula used for discriminant is

  $B^2-4AC$

Calculation:

given the following equation:

  $3x^2-5xy+2y^2-7x-14y=-1$

Comparing with the standard equation for a Quadratic Curve,

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Thus,

  $A=3,B=-5,C=2$

Now, discriminant $D=B^2-4AC$

  $\begin{array}{l}={\left(-5\right)}^2-4\left(3\right)\left(2\right)\\ =1\end{array}$

Since $D>0$ , the given curve represents a hyperbola.

Conclusion:

Since $D>0$ , the given curve represents a hyperbola.