# Question # 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.

Conic sections
ANSWERED Determine whether the statement If $$\displaystyle{D}\ne{0}{\quad\text{or}\quad}{E}\ne{0}$$,
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. 2021-02-16
Step 1
We have given a statement:
If $$\displaystyle{D}\ne{0}{\quad\text{or}\quad}{E}\ne{0}$$,
then graph of $$\displaystyle{y}^{2}-{x}^{2}+{D}{x}+{E}{y}={0}$$ is a hyperbolais.
Step 2
We know the general form of conic section:
$$\displaystyle{A}{x}^{2}+{B}{x}{y}+{C}{y}^{2}+{D}{x}+{E}{y}+{F}={0}$$
To find the type of conic section we solve for $$\displaystyle{B}^{2}-{4}{A}{C}:$$
(i) If $$\displaystyle{B}^{2}-{4}{A}{C}<{0}$$</span> then the conic section is ellipse.
(ii) If $$\displaystyle{B}^{2}-{4}{A}{C}<{0}{\quad\text{and}\quad}{A}={C},{B}={0}$$</span> then we have a perfect circle.
(iii) If $$\displaystyle{B}^{2}-{4}{A}{C}={0}$$, then we have a parabola.
$$\displaystyle{B}^{2}-{4}{A}{C}>{0}$$, then we have a hyperbola
Hyperbola defined as:
$$\displaystyle\frac{{{\left({x}-{h}\right)}^{2}}}{{a}^{2}}+\frac{{{\left({y}-{k}\right)}^{2}}}{{b}^{2}}={1}$$
Where (h, k) are center.
When $$\displaystyle{E},{D}={0},\text{then}{\left({h},{k}\right)}={\left({0},{0}\right)}$$
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.