The question asks:
Find the equation of the hyperboloid shown in the figure.
Now, given that the equation for an one-sheet hyperboloid follows the format
$$(\frac{x}{a})^2+(\frac{y}{b})^2=(\frac{z}{c})^2+1$$
I was able to solve for $a$ and $b$, but not for $c$.
$$(\frac{x}{2})^2+(\frac{y}{6})^2=(\frac{z}{c})^2+1$$
What am I missing? Thank you for your consideration.
Best Answer
Since in the equation we have three parameters, three points on the surface are sufficient to find the equation.
You have just find two parameters so you need only one point more, e.g. $(4,0,9)$ from the graph. With these coordinates in the equation we find: $$ \left(\frac{4}{2} \right)^2=\left(\frac{9}{c} \right)^2+1 \Rightarrow c^2=27 $$ so the equation is: $$ \frac{x^2}{4} +\frac{y^2}{36}-\frac{z^2}{27}=1 $$ and note that it is satisfied also for the coordinates of the other point in the graph $(0,12,9)$