Question

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.

Answer 1

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.