Question

# Find the required information and graph: displaystyle{2}{x}^{2}+{2}{y}^{2}+{2}{x}+{14}{y}+{17}={0}

Conic sections
Find the required information and graph:
$$\displaystyle{2}{x}^{2}+{2}{y}^{2}+{2}{x}+{14}{y}+{17}={0}$$

2021-02-26
Step 1
Consider the provided equation,
$$\displaystyle{2}{x}^{2}+{2}{y}^{2}+{2}{x}+{14}{y}+{17}={0}$$
Classify the conic section and find the center.
We can write as,
$$\displaystyle{2}{x}^{2}+{2}{x}+{2}{y}^{2}+{14}{y}+{17}={0}$$
$$\displaystyle{2}{\left({x}^{2}+{x}\right)}+{2}{\left({y}^{2}+{7}{y}\right)}=-{17}$$
$$\displaystyle{\left({x}^{2}+{x}\right)}+{\left({y}^{2}+{7}{y}\right)}=-\frac{17}{{2}}$$
$$\displaystyle{\left({x}^{2}+{x}+\frac{1}{{4}}\right)}+{\left({y}^{2}+{7}{y}+\frac{49}{{4}}\right)}=-\frac{17}{{2}}+\frac{49}{{4}}+\frac{1}{{4}}$$
Step 2
Simplifying further,
$$\displaystyle{\left({x}+\frac{1}{{2}}\right)}^{2}+{\left({y}+\frac{7}{{2}}\right)}^{2}={4}$$
$$\displaystyle{\left({x}-{\left(-\frac{1}{{2}}\right)}\right)}^{2}+{\left({y}-{\left(-\frac{7}{{2}}\right)}\right)}^{2}={2}^{2}$$
$$\displaystyle{\left({x}-{a}\right)}^{2}+{\left({y}-{b}\right)}^{2}={r}^{2}$$ is the circle equation with radius r, centered at (a, b).
Thus, this is a circle.
So, the center or the circle $$\displaystyle{\left(-\frac{1}{{2}},-\frac{7}{{2}}\right)}.$$