what is the maximum number of roots of quadratic function with 3 variables?

Question

Given the general quadratic form with $3$ variables $(x,y,z):ax^2 + by^2 +cz^2 + dxy + eyz +fzx + gx + hy + iz = 0$ satisfies $x^2 + y^2 + z^2 = 1$

I would like to ask what is the maximum number of roots of the provided equation.

Thanks in advance

It seems obvious from the first few comments that the set of solution which is the intersection between the two surface is infinity.

If adding 2 constraints: 1. the solutions are real numbers and 2. any 4 points in the set of solution can not lie on the same plane, what is maximum number of solutions?

Answer 1

The question could be written more precisely: when you say "satisfies $x^2+y^2+z^2=1$", I assume you mean that you are looking for simulateneous solutions of both equations.

Anyway, the answer is that such a system of equations can have infinitely many solutions. Geometrically, each equation defines a surface in 3-dimensional space, and their common solutions are the intersection points of those surfaces.

If we are talking about real-number solutions, then your second equation defines the unit sphere, and it's not hard to visualise another surface (the solution set of the first equation) intersecting it in a curve, say.

Answer 2

There could be infinitely many solutions to those two equations. Let $a=b=c=d=e=f=g=h=0$ and let $i=1$.

Then the first equation becomes $z=0$, which is the $xy$-plane, and that intersects the unit sphere in the unit circle.