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.

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.