Question

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

Conic sections
ANSWERED
asked 2021-02-25
Find the required information and graph:
\(\displaystyle{2}{x}^{2}+{2}{y}^{2}+{2}{x}+{14}{y}+{17}={0}\)

Answers (1)

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