Chapter A5.3, Problem 1E

To determine

Whether the given equation represents parabola, ellipse, or hyperbola.

Answer to Problem 1E

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

Explanation of Solution

Given:

  $x^2-3xy+y^2-x=0$

Formula Used:

  $B^2-4AC$

Calculation:

We have been given the following equation:

  $x^2-3xy+y^2-x=0$

Comparing with the standard equation for a Quadratic Curve,

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

We get:

  $A=1,B=-3,C=1$

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

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

Since $D>0$ , the given curve represents a $\boxed{hyperbola}$

Conclusion:

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