# How do you graph 9 x² + y² + 18x - 6y + 9 =0?

May 15, 2018

#### Explanation:

General equation of an ellipse is of the form ${\left(x - h\right)}^{2} / {a}^{2} + {\left(y - k\right)}^{2} / {b}^{2} = 1$, where $\left(h , k\right)$ is the center of ellipse and axis are $2 a$ and $2 b$, with larger one as major axis an other minor axis. We can also find vertices by adding $\pm a$ to $h$ (keeping ordinate same) and $\pm b$ to $k$ (keeping abscissa same).

We can write the equation $9 {x}^{2} + {y}^{2} + 18 x - 6 y + 9 = 0$ as

$9 \left({x}^{2} + 2 x + 1\right) + \left({y}^{2} - 6 y + 9\right) = 9 + 9 - 9$

or $9 {\left(x + 1\right)}^{2} + {\left(y - 3\right)}^{2} = 9$

or ${\left(x + 1\right)}^{2} / 1 + {\left(y - 3\right)}^{2} / 9 = 1$

or ${\left(x + 1\right)}^{2} / {1}^{2} + {\left(y - 3\right)}^{2} / {3}^{2} = 1$

Hence center of ellipse is $\left(- 1 , 3\right)$, while major axis parallel to $y$-axis is $2 \times 3 = 6$ and minor axis parallel to $x$-axis is $2 \times 1 = 2$.

Hence vertices are $\left(0 , 3\right)$, $\left(- 2 , 3\right)$, $\left(- 1 , 0\right)$ and $\left(- 1 , 6\right)$.

graph{9x^2+y^2+18x-6y+9=0 [-11.38, 8.62, -2.12, 7.88]}