Chapter A5.3, Problem 2E

To determine

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

Answer to Problem 2E

Since $D=0$ , the given curve represents a parabola.

Explanation of Solution

Given:

  $3x^2-18xy+27y^2-5x+7y=-4$

Formula Used:

  $B^2-4AC$

Calculation:

We have been given the following equation:

  $3x^2-18xy+27y^2-5x+7y=-4$

Comparing with the standard equation for a Quadratic Curve,

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

We get:

  $A=3,B=-18,C=27$

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

  $\begin{array}{l}={\left(-18\right)}^2-4\left(3\right)\left(27\right)\\ =0\end{array}$

Since $D=0$ , the given curve represents a $\boxed{parabola}$

Conclusion:

Since $D=0$ , the given curve represents a parabola.