Answer
An elliptic cone
Work Step by Step
The equation
$$14z^2=3x^2+7y^2$$ can be rewritten as follows
$$
\left(\frac{z}{1/\sqrt {14}}\right)^{2}=\left(\frac{x}{1/\sqrt 3}\right)^{2}+\left(\frac{y}{1/\sqrt{7}}\right)^{2}
$$ which is an equation of an elliptic cone with $ a=1/\sqrt{ 3}, b=1/\sqrt{7}, c=1/\sqrt{ 14}$.
(See equations on page 691.)
