Chapter A5.2, Problem 9E

To determine

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

Answer to Problem 9E

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

Explanation of Solution

Given:

  $xy+y^2-3x=5$

Formula Used:

The formula used for discriminant is

  $B^2-4AC$

Calculation:

Given the following equation:

  $xy+y^2-3x=5$

Comparing with the standard equation for a Quadratic Curve,

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Thus,

  $A=0,B=1,C=1$

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

  $\begin{array}{l}={\left(1\right)}^2-4\left(0\right)\left(1\right)\\ =1\end{array}$

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

Conclusion:

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