a first degree linear equation $ax+by+c=0$ represents a family of straight lines passing through a fixed point if and only if there is linear relationship between a,b and c? How can we prove this? Can the relation between a,b,c be a quadratic one or cubic one and so on or does it always have to be linear and why?
I tried to take any relation between a,b,c and put it in the equation of line but it turns out nothing can be said about whether it represents a family of lines or not?
Hint:
Suppose, for example, that the fixed point is $(2,3)$. Then no matter what the values of $a$, $b$, and $c$ are, the equation $2a + 3b = c$ is always true. This is a linear relationship between $a$, $b$, and $c$.
And note that there's nothing special about the point $(2,3)$. I just chose it to have a concrete example to explain the concept. You can run through the exact same argument and use $(x_0, y_0)$ to represent the coordinates of an arbitrary fixed point. This will basically give you one direction of the proof. Technically both directions if you're careful with how you choose your words.