I need to find conditions for the existence of non-trivial solutions to a multivariable polynomial system in two cases:

**The first case:**

$f1: a_1x^2+a_2xy+a_3y^2+a_4z^2=0$

$f2: b_1x^2+b_2xy+b_3y^2+b_4z^2=0$

**The second case:**

$f3: a_1x^2+a_2xy+a_3y^2+a_3z^2+a_5xz=0$

$f4: b_1x^2+b_2xy+b_3y^2+b_3z^2+b_5xz=0$

(note that the coefficient of y2 and z2 are equal)

All the coefficient are reals number and the variable x,y,z should be real too.

I tried to use resultant and eliminating one variable, for even a more simple case:

$f5: a_1x^2+a_2xy+a_3y^2+a_3z^2=0$

$f6: b_1x^2+b_2xy+b_3y^2+b_3z^2=0$

The problem that i get different solution:

1: if I try to eliminate $z$, I get $RES(f5,f6,z)=x^2[(a_2c_1−a_1c_2)x+2(b_2c_1−b_1c_2)y]^2$ So there is always non trivial (x≠0 or y≠0) real x,y such that RES(f5,f6,z)=0

2: On the other hand if I eliminte $x$ I get: $ \small RES(f5,f6,x)=[(a_2c_1−a_1c_2)^2+4(a_2b_1−a_1b_2)(b_2c_1−b_1c_2)]x^4+4(b_2c_1−b_1c_2)^2x^2z^2$

thus a non-trivial solution exist only if $(a_2c_1−a_1c_2)^2+4(a_2b_1−a_1b_2)(b_2c_1−b_1c_2)≤0$.

I tried to check this numerically and it seems like the second condition is the right one, so why the first attempt eliminating z is wrong?

Any help will be appreciated