Find the center, foci, vertices, asymptotes, and radius, if necessary, of the conic sections in the equation: displaystyle{x}^{2}+{4}{x}+{y}^{2}={12}

Question
Conic sections
Find the center, foci, vertices, asymptotes, and radius, if necessary, of the conic sections in the equation:
$$\displaystyle{x}^{2}+{4}{x}+{y}^{2}={12}$$

2021-02-01
Since you have submitted two questions, we'll answer the first question. For the second question please submit the question again and specify it.
$$\displaystyle{x}^{2}+{4}{x}+{y}^{2}={12}$$ is a circle
First write the equation in standard form of circle
$$\displaystyle{x}^{2}+{4}{x}+{y}^{2}={12}$$
$$\displaystyle{x}^{2}+{4}{x}+{4}+{y}^{2}={12}+{4}$$
$$\displaystyle{\left({x}+{2}\right)}^{2}+{y}^{2}={16}$$
Compare the circle with the standard form of circle and find h,k,r
$$\displaystyle{\left({x}+{2}\right)}^{2}+{y}^{2}={16}$$
$$\displaystyle{\left({x}+{2}\right)}^{2}+{\left({y}-{0}\right)}^{2}={16}$$
$$\displaystyle{\left({x}-{h}\right)}^{2}+{\left({y}-{k}\right)}^{2}={r}^{2}$$
$$\displaystyle{h}=-{2},{k}={0}$$
$$\displaystyle{r}^{2}={16}$$
$$\displaystyle{r}={4}$$
Plug h,k,r in the formula of center and radius and find them.
Center $$\displaystyle={\left({h},{k}\right)}={\left(-{2},{0}\right)}$$
Radius $$\displaystyle={r}={4}$$ units
Answer: Center $$\displaystyle={\left(-{2},{0}\right)}$$
Radius $$\displaystyle={4}$$ units

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$$\displaystyle{\left({a}\right)}{4}{x}{2}-{9}{y}{2}={12}{\left({b}\right)}-{4}{x}+{9}{y}{2}={0}$$
$$\displaystyle{\left({c}\right)}{4}{y}{2}+{9}{x}{2}={12}{\left({d}\right)}{4}{x}{3}+{9}{y}{3}={12}$$