Step 1

We need to determine the shape of the given conic section using its equation.

\(\displaystyle{x}^{2}-{2}{x}{y}+{y}^{2}+{x}-{2}{y}-{1}={0}\)

Step 2

We know that, for the general equation of conic section

\(\displaystyle{A}{x}^{2}+{B}{x}{y}+{C}{y}^{2}+{D}{x}+{E}{y}+{F}={0}\)

if \(\displaystyle{B}^{2}-{4}{A}{C}={0},\) then the conic must be a parabola.

For the given equation, \(\displaystyle{A}={1},{B}=-{2},{C}={1}\)

\(\displaystyle{B}^{2}-{4}{A}{C}={\left(-{2}\right)}^{2}-{4}{\left({1}\right)}{\left({1}\right)}\)

\(\displaystyle={0}=>\) given conic section is a parabola

Step 3

The graph of the parabola has been shown below.

We need to determine the shape of the given conic section using its equation.

\(\displaystyle{x}^{2}-{2}{x}{y}+{y}^{2}+{x}-{2}{y}-{1}={0}\)

Step 2

We know that, for the general equation of conic section

\(\displaystyle{A}{x}^{2}+{B}{x}{y}+{C}{y}^{2}+{D}{x}+{E}{y}+{F}={0}\)

if \(\displaystyle{B}^{2}-{4}{A}{C}={0},\) then the conic must be a parabola.

For the given equation, \(\displaystyle{A}={1},{B}=-{2},{C}={1}\)

\(\displaystyle{B}^{2}-{4}{A}{C}={\left(-{2}\right)}^{2}-{4}{\left({1}\right)}{\left({1}\right)}\)

\(\displaystyle={0}=>\) given conic section is a parabola

Step 3

The graph of the parabola has been shown below.