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

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

### Relevant Questions

For Exercise, an equation of a degenerate conic section is given. Complete the square and describe the graph of each equation.
$$\displaystyle{9}{x}{2}+{4}{y}{2}-{24}{y}+{36}={0}$$
Given the following conic section, determine its shape and then sketch its graph.
$$\displaystyle{x}^{2}-{2}{x}{y}+{y}^{2}+{x}-{2}{y}-{1}={0}$$
The exercise provides the equation for a degenerate conical section. Fill in the square and describe the graph of each equation.
$$\displaystyle{4}{x}{2}-{y}{2}-{32}{x}-{4}{y}+{60}={0}$$
Write the following equation in standard form and sketch its graph
$$\displaystyle{9}{x}^{2}+{72}{x}-{64}{y}^{2}+{128}{y}+{80}={0}$$
Consider the equation:
$$\displaystyle{2}\sqrt{{3}}{x}^{2}-{6}{x}{y}+\sqrt{{3}}{x}+{3}{y}={0}$$
a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola?
b) Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies $$\displaystyle{0}\le\varphi\le\pi\text{/}{2}$$)
Determine whether the statement If $$\displaystyle{D}\ne{0}{\quad\text{or}\quad}{E}\ne{0}$$,
then the graph of $$\displaystyle{y}^{2}-{x}^{2}+{D}{x}+{E}{y}={0}$$ is a hyperbolais true or false. If it is false, explain why or give an example that shows it is false.
Solve for the equation in standard form of the following conic sections and graph the curve on a Cartesian plane indicating important points.
1. An ellipse passing through (4, 3) and (6, 2)
2. A parabola with axis parallel to the x-axis and passing through (5, 4), (11, -2) and (21, -4)
3. The hyperbola given by $$\displaystyle{5}{x}^{2}-{4}{y}^{2}={20}{x}+{24}{y}+{36}.$$
$$\displaystyle{2}{x}^{2}+{2}{y}^{2}-{8}{x}-{8}{y}={0}$$
$$\displaystyle{3}{y}^{2}+{8}{x}+{24}{y}+{40}={0}$$
(a) Given the conic section $$\displaystyle{r}=\frac{5}{{{7}+{3} \cos{{\left(\theta\right)}}}}$$, find the x and y intercept(s) and the focus(foci).
(b) Given the conic section $$\displaystyle{r}=\frac{5}{{{2}+{5} \sin{{\left(\theta\right)}}}}$$, find the x and y intercept(s) and the focus(foci).