# Given the following conic section, determine its shape and then sketch its graph. displaystyle{x}^{2}-{2}{x}{y}+{y}^{2}+{x}-{2}{y}-{1}={0}

Question
Conic sections
Given the following conic section, determine its shape and then sketch its graph.
$$\displaystyle{x}^{2}-{2}{x}{y}+{y}^{2}+{x}-{2}{y}-{1}={0}$$

2021-02-23
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.

### Relevant Questions

For Exercise, an equation of a degenerate conic section is given. Complete the square and describe the graph of each equation.
$$\displaystyle{9}{x}{2}+{4}{y}{2}-{24}{y}+{36}={0}$$
Write the following equation in standard form and sketch its graph
$$\displaystyle{9}{x}^{2}+{72}{x}-{64}{y}^{2}+{128}{y}+{80}={0}$$
Identify the conic section given by $$\displaystyle{y}^{2}+{2}{y}={4}{x}^{2}+{3}$$
Find its $$\frac{\text{vertex}}{\text{vertices}}\ \text{and}\ \frac{\text{focus}}{\text{foci}}$$
Identify the conic section given by the polar equation $$\displaystyle{r}=\frac{4}{{{1}- \cos{\theta}}}$$ and also determine its directrix.
Solve for the equation in standard form of the following conic sections and graph the curve on a Cartesian plane indicating important points.
1. An ellipse passing through (4, 3) and (6, 2)
2. A parabola with axis parallel to the x-axis and passing through (5, 4), (11, -2) and (21, -4)
3. The hyperbola given by $$\displaystyle{5}{x}^{2}-{4}{y}^{2}={20}{x}+{24}{y}+{36}.$$
Identify the type of conic section whose equation is given.
Given:
$$\displaystyle{2}{x}^{2}={y}^{2}+{2}$$
To determine: The type of conics.
Find out what kind of conic section the following quadratic form represents and transform it to principal axes. Express $$\displaystyle\vec{{x}}^{T}={\left[{x}_{{1}}{x}_{{2}}\right]}$$ in terms of the new coordinate vector $$\displaystyle\vec{{y}}^{T}={\left[{y}_{{1}}{y}_{{2}}\right]}$$
$$\displaystyle{{x}_{{1}}^{{2}}}-{12}{x}_{{1}}{x}_{{2}}+{{x}_{{2}}^{{2}}}={70}$$
A latus rectum of a conic section is a chord through a focus parallel to the directrix. Find the area bounded by the parabola $$\displaystyle{y}={x}^{2}\text{/}{\left({4}{c}\right)}$$ and its latus rectum.
$$\displaystyle{2}{x}{2}-{8}{x}{y}+{3}{y}{2}-{4}={0}$$
Determine the equation of a conic section...(Hyperbola) Given: center (-9, 1) distance between $$\displaystyle{F}{1}{\quad\text{and}\quad}{F}{2}={20}$$ units
distance between $$\displaystyle{C}{V}{1}{\quad\text{and}\quad}{C}{V}{2}={4}$$ units orientation = vertical