Reduce the equation to standard form, classify the surface and sketch it.

Question

Reduce the equation to standard form, classify the surface and sketch it.

4y² + z² - x - 16y - 4z + 20 = 0

Answer 1

z²-4z +4y²-16y -x = -20

(z-2)²+4(y-2)²-x = -60+4+16

(z-2)²/4 + (y-2)²+10 = x/4

it is a paraboloid with the vertex at (40, 2, 2) that opens toward the positive x-direction. Traces of it with planes parallel to the y-z plane are ellipses of the form (x-2)²/k + (y-2)²/(k/4) = 1, where k> 0 increases as x increases.

Answer 2

4y² + z² - x - 16y - 4z + 20 = 0

(-x) + (4y² - 16y) + (z² -4z) = -20 <===== Combine link terms, and bring the constant to the other side

(-x) + (4y² - 16y + 16) + (z² - 4z + 4) = -20 + 16 + 4 <===== Using "busting the B" to be able to change form into perfect squares.

(-x) + (2y - 4)² + (z - 2)² = 0 <===== Standard form

To summarize: We are combining like terms. Then add constants to both sides of the equation to allow for our variables to have perfect squares. From there we can simplify to our standard form.

Graph (Classified as a paraboloid):

https://www.flickr.com/photos/195092992@N08/51883936493/in/dateposted-public/

(If you do not want to use the link you can use an online grapher. I recommend GeoGebra)

Origin at: (0, 2, 2)