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 \(\displaystyle{\left(-\frac{1}{{2}},-\frac{7}{{2}}\right)}.\)

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 \(\displaystyle{\left(-\frac{1}{{2}},-\frac{7}{{2}}\right)}.\)