What is the Surface of the Equation S88 and How Can it be Graphed Online?

[karush]

$ \tiny{231.14.88}\\$
$\textsf{Identify and briefly describe the surface of the equation}\\$
\begin{align*}
S_{88}&=\frac{-x^2-y^2+z^2}{9+6x-8y}=26
\end{align*}
$\textit{this had no template example but on}\\$
$\textit{ W|A it looked like a torus?}\\$
$\textit{where is a good online 3d graphing calculator}$

 

[MarkFL]

Multiplying through by $9+6x-8y$ and distributing the $26$, we obtain

$$-x^2-y^2+z^2=156x-208y+234$$

$$-x^2-156x-y^2+208y+z^2=234$$

$$-\left(x^2+156x\right)-\left(y^2-208y\right)+z^2=234$$

$$-\left(x^2+156x+6084\right)-\left(y^2-208y+10816\right)+z^2=234-6084-10816$$

$$-(x+78)^2-(y-104)^2+z^2=-16666$$

$$(x+78)^2+(y-104)^2-z^2=16666$$

$$\frac{(x+78)^2}{16666}+\frac{(y-104)^2}{16666}-\frac{z^2}{16666}=1$$

Thus, we see this is a hyperboloid of one sheet. :D

 

[karush]

thank you that was very helpful
new stuff for me.

 

[HallsofIvy]

Strictly speaking, it is the set of points on that hyperboloid that do not satisfy 9+ 6x- 8y= 0.