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

1 Answer
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...
(16x^2 + 32x)+(y^2-18y)=119

Now take out the factor of 16 in the x "group".
16(x^2 + 2x) + (y^2-18y)=119

Next, complete the squares
16(x^2+2x+1)+(y^2-18y+81)=119+16+81
16(x+1)^2+(y-9)^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:

(x-h)^2/a+(y-k)^2/b = 1

So, let's get this formula into that form.
(x+1)^2/13.5 + (y-9)^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 sqrt13.5 or about 3.67, and the vertical axis (also the major axis of this ellipse) will have a length of sqrt216 (or 6sqrt6), 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]}