# 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.

Determine whether the statement If $D\ne 0\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}E\ne 0$,
then the graph of ${y}^{2}-{x}^{2}+Dx+Ey=0$ is a hyperbolais true or false. If it is false, explain why or give an example that shows it is false.
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Step 1
We have given a statement:
If $D\ne 0\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}E\ne 0$,
then graph of ${y}^{2}-{x}^{2}+Dx+Ey=0$ is a hyperbolais.
Step 2
We know the general form of conic section:
$A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$
To find the type of conic section we solve for ${B}^{2}-4AC:$
(i) If ${B}^{2}-4AC<0$ then the conic section is ellipse.
(ii) If ${B}^{2}-4AC<0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}A=C,B=0$ then we have a perfect circle.
(iii) If ${B}^{2}-4AC=0$, then we have a parabola.
${B}^{2}-4AC>0$, then we have a hyperbola
Hyperbola defined as:
$\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1$
Where (h, k) are center.
When $E,D=0,\text{then}\left(h,k\right)=\left(0,0\right)$
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.

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