# 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}

For Exercise, an equation of a degenerate conic section is given. Complete the square and describe the graph of each equation.
$$\displaystyle{9}{x}^{2}+{4}{y}^{2}-{24}{y}+{36}={0}$$

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SchulzD
Step 1
The given equation is $$\displaystyle{9}{x}^{2}+{4}{y}^{2}-{24}{y}+{36}={0}.$$
Use completing the square method and obtain the standard form
Step 2
$$\displaystyle{4}{y}^{2}-{24}{y}=-{9}{x}^{2}-{36}$$
$$\displaystyle{4}{\left({y}^{2}-{6}{y}\right)}=-{9}{x}^{2}-{36}$$
$$\displaystyle{4}{\left({y}^{2}-{6}{y}+{3}^{2}-{3}^{2}\right)}=-{9}{x}^{2}-{36}$$
$$\displaystyle{4}{\left({y}+{3}\right)}^{2}-{36}=-{9}{x}^{2}-{36}$$
$$\displaystyle{4}{\left({y}+{3}\right)}^{2}=-{9}{x}^{2}$$
$$\displaystyle{\left({y}+{3}\right)}^{2}=-{\left(\frac{3}{{2}}{x}\right)}^{2}$$
$$\displaystyle{y}+{3}=-\frac{3}{{2}}{x}$$
$$\displaystyle{2}{y}+{6}=-{3}{x}$$
$$\displaystyle{4}{x}+{2}{y}+{6}={0}$$
Step 3
Draw the graph of $$\displaystyle{4}{x}+{2}{y}+{6}={0}$$