# How do I graph 16x^2+y^2+32x-18y=119 algebraically?

Sep 2, 2015

Get the equation into a familiar form, and then figure out what each number in that equation means.

#### Explanation:

This looks like the equation of a circle. The best way to get these into a graphable form is to play around with the equation and complete squares. Let's first regroup these...
$\left(16 {x}^{2} + 32 x\right) + \left({y}^{2} - 18 y\right) = 119$

Now take out the factor of 16 in the x "group".
$16 \left({x}^{2} + 2 x\right) + \left({y}^{2} - 18 y\right) = 119$

Next, complete the squares
$16 \left({x}^{2} + 2 x + 1\right) + \left({y}^{2} - 18 y + 81\right) = 119 + 16 + 81$
$16 {\left(x + 1\right)}^{2} + {\left(y - 9\right)}^{2} = 216$

Hmm... this would be the equation of a circle, except there's a factor of 16 in front of the x group. That means it must be an ellipse.
An ellipse with center (h, k) and a horizontal axis "a" and vertical axis "b" (regardless of which one is the major axis) is as follows:

${\left(x - h\right)}^{2} / a + {\left(y - k\right)}^{2} / b = 1$

So, let's get this formula into that form.
${\left(x + 1\right)}^{2} / 13.5 + {\left(y - 9\right)}^{2} / 216 = 1$ (Divide by 216) That's it!

So, this ellipse is going to be centered at (-1, 9). Also, the horizontal axis will have a length of $\sqrt{13.5}$ or about $3.67$, and the vertical axis (also the major axis of this ellipse) will have a length of $\sqrt{216}$ (or $6 \sqrt{6}$), or about $14.7$.

If you were to graph this by hand, you would draw a dot at (-1, 9), draw a horizontal line extending about 3.67 units on either side of the dot, and a vertical line extending about 4.7 units on either side of the dot. Then, draw an oval connecting the tips of the four lines.

If this doesn't make sense, here's a graph of the ellipse.
graph{16x^2 + y^2+32x-18y =119 [-34.86, 32.84, -8, 25.84]}