Chapter A5.2, Problem 4E

To determine

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

Answer to Problem 4E

Since $D<0$ , the given curve represents a ellipse.

Explanation of Solution

Given:

  $2x^2-\sqrt{15xy}+2y^2+x+y=0$

Formula Used:

The formula used for discriminant is

  $B^2-4AC$

Calculation:

Given the following equation:

  $2x^2-\sqrt{15xy}+2y^2+x+y=0$

Comparing with the standard equation for a Quadratic Curve,

  $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$

Thus,

  $A=2,B=-\sqrt{15},C=2$

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

  $\begin{array}{l}={\left(-\sqrt{15}\right)}^2-4\left(2\right)\left(2\right)\\ =-1\end{array}$

Since $D<0$ , the given curve represents a ellipse.

Conclusion:

Since $D<0$ , the given curve represents a ellipse.