# The conic for the equation displaystyle{left({x}+{2}right)}^{2}+{left({y}-{1}right)}^{2}={4} and also describe the translation of the conic from the standard position.

Question
Conic sections
The conic for the equation $$\displaystyle{\left({x}+{2}\right)}^{2}+{\left({y}-{1}\right)}^{2}={4}$$ and also describe the translation of the conic from the standard position.

2021-02-12
Consider the equation,
$$\displaystyle{\left({x}+{2}\right)}^{2}+{\left({y}-{1}\right)}^{2}={4}$$
The above equation is also written as follows,
$$\displaystyle{\left({x}+{2}\right)}^{2}+{\left({y}-{1}\right)}^{2}={2}^{2}$$
Now compare the above equation with the standard form of the equation of the circle, that is $$\displaystyle{\left({x}-{h}\right)}^{2}+{\left({y}-{k}\right)}^{2}={r}^{2}.$$
So,
$$\displaystyle{h}=-{2},{k}={1}{\quad\text{and}\quad}{r}={2}$$
Thus, the equation $$\displaystyle{\left({x}+{2}\right)}^{2}+{\left({y}-{1}\right)}^{2}={4}$$ is the equation of circle with center at (-2, 1)
and radius $$\displaystyle{r}={2}$$ as shown below,
Therefore, the graph has been shifted 1 unit upward and 2 units to the left from the standard position.

### Relevant Questions

The conic for the equation $$\displaystyle{\left({x}+{1}\right)}^{2}={4}{\left(-{1}\right)}{\left({y}-{2}\right)}$$ and also describe the translation of the from standard position.
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 $$\displaystyle{5}{x}^{2}-{4}{y}^{2}={20}{x}+{24}{y}+{36}.$$
For Exercise, an equation of a degenerate conic section is given. Complete the square and describe the graph of each equation.
$$\displaystyle{9}{x}{2}+{4}{y}{2}-{24}{y}+{36}={0}$$
Decide if the equation defines an ellipse, a hyperbola, a parabola, or no conic section at all.
$$\displaystyle{\left({a}\right)}{4}{x}{2}-{9}{y}{2}={12}{\left({b}\right)}-{4}{x}+{9}{y}{2}={0}$$
$$\displaystyle{\left({c}\right)}{4}{y}{2}+{9}{x}{2}={12}{\left({d}\right)}{4}{x}{3}+{9}{y}{3}={12}$$
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.
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 $$\displaystyle\vec{{y}}^{T}={\left[{y}_{{1}}{y}_{{2}}\right]}$$
$$\displaystyle{{x}_{{1}}^{{2}}}-{12}{x}_{{1}}{x}_{{2}}+{{x}_{{2}}^{{2}}}={70}$$
$$\displaystyle{4}{x}{2}-{y}{2}-{32}{x}-{4}{y}+{60}={0}$$
$$\displaystyle{x}^{2}+{4}{x}+{y}^{2}={12}$$
Identify the conic section given by the polar equation $$\displaystyle{r}=\frac{4}{{{1}- \cos{\theta}}}$$ and also determine its directrix.
Solve, a.Determine the conic section of the polar equation $$\displaystyle{r}=\frac{8}{{{2}+{2} \sin{\theta}}}$$ represents. b. Describe the location of a directrix from the focus located at the pole.