I am given the equation: $9x^2 + 4y^2 + z = 3$
The standard equation of a paraboloid parallel to z-axis is: $$\frac{z-z_0}{c} = \frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2}$$ I think the given equation should be a paraboloid due to the $z$ component, then I try to put it in a std. form.
$9x^2+ 4y^2 + z = 3$
$\frac{9x^2}{3} + \frac{4y^2}{3} + \frac{z}{3} = 1$
$3x^2 + \frac{4}{3}y^2 = 1 - \frac{z}{3}$
And then, I am confused since this is obviously not the std form. The one and also the negative value of z is not std.
I would appreciate any tips or someone pointing me the right way!
Note $1 - \frac{z}{3} = \frac{3 - z}{3} = \frac{z - 3}{-3}$. Thus, $z_0 = 3$ and $c = -3$, along with $x_0 = y_0 = 0$, $a = \frac{1}{\sqrt{3}}$ and $b = \frac{\sqrt{3}}{2}$, allows your equation to be in standard form.