Is this family of lines just one line?

Question

Question:

If $a,c$ and $b$ are in G.P. with common ratio $2$, then the family of lines $ax+by+c=0$ always passes through point

$A) (-1,2)$

$B) (2,-1)$

$C) (1,2)$

$D)$ none of these

My Attempt:

$c=2a, b=4a$, so,

$ax+4ay+2a=0\implies x+4y+2=0$

Does that mean the given family of lines is just one line?

Answer 1

A family is a function, here $$\mathbb R^*\to \mathcal D$$ $$a\mapsto \{(x,y)\in \Bbb R^2: ax+4ay+2a=0\}$$where $\mathcal D$ is the set of the lines of $\mathbb R^2$.

But $\forall a\in \mathbb R^*, \{(x,y)\in \Bbb R^2: ax+4ay+2a=0\}=\{(x,y)\in \Bbb R^2: a(x+4y+2)=0\}=\{(x,y)\in \mathbb R^2:x+4y+2=0\}$

So, yes the family is the constant function equal to $D:x+4y+2=0$, which goes through $(2,-1)$.