Question

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

Answer 1

Please see below.

Explanation:

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

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

`9(x^2+2x+1)+(y^2-6y+9)=9+9-9`

or `9(x+1)^2+(y-3)^2=9`

or `(x+1)^2/1+(y-3)^2/9=1`

or `(x+1)^2/1^2+(y-3)^2/3^2=1`

Hence center of ellipse is `(-1,3)`, while major axis parallel to `y`-axis is `2xx3=6` and minor axis parallel to `x`-axis is `2xx1=2`.

Hence vertices are `(0,3)`, `(-2,3)`, `(-1,0)` and `(-1,6)`.

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