A specific case of quadratic forms

Question

I have a quadric as follows:

$$ax^2+by^2+bz^2+yz=0.$$ I am curious to know which shapes in $\mathbb{R}^3$ this equation describes for different value of $a$ and $b$?

Answer 1

if $b\ge \frac 12$, then there isn't enough room to move about.

$a^2 x^2, b^2 y^2, b^2 z^2,$ are all greater than $0.$

$b^2 y^2 + b^2 z^2 > yz.$

And the only solution is $(0,0,0).$

If $|b| < \frac 12 $, then you get a double cone.