Q:

What is a linear condition on the coeffcients of a quadric?

Suppose that we have a quadric $ Q. $ What does it mean to say that the requirement that $ Q = \lbrace f = 0 \rbrace $ passes through a given point is a linear condition on the coefficients of $ f$?
My apologies if this is quite basic. I feel that this is mostly a matter of terminology.

A:

Question: does $$  a x^2 +  b xy + c y^2 = 0  $$ pass through $$ x=7, y=-2 \; ? $$
Answer: does
$$  49 a - 14 b + 4 c = 0 \; ? $$

