Determine whether the statement If displaystyle{D}ne{0}{quadtext{or}quad}{E}ne{0}, then the graph of displaystyle{y}^{2}-{x}^{2}+{D}{x}+{E}{y}={0} is a hyperbolais true or false. If it is false, explain why or give an example that shows it is false.

Step 1
We have given a statement:
If ${\displaystyle D\ne 0\text{or}E\ne 0}$,
then graph of ${\displaystyle y^2-x^2+Dx+Ey=0}$ is a hyperbolais.
Step 2
We know the general form of conic section:
${\displaystyle A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0}$
To find the type of conic section we solve for ${\displaystyle B^2-4AC:}$
(i) If ${\displaystyle B^2-4AC<0}$ then the conic section is ellipse.
(ii) If ${\displaystyle B^2-4AC<0\text{and}A=C,B=0}$ then we have a perfect circle.
(iii) If ${\displaystyle B^2-4AC=0}$, then we have a parabola.
${\displaystyle B^2-4AC>0}$, then we have a hyperbola
Hyperbola defined as:
${\displaystyle \frac{{(x-h)}^2}{a^2}+\frac{{(y-k)}^2}{b^2}=1}$
Where (h, k) are center.
When ${\displaystyle E,D=0,\text{then}(h,k)=(0,0)}$
Then the center will be at (0,0)
Step 3
Hence, the given statement is incorrect since the given condition is not mandatory for a conic section to be a hyperbola.