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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?

Matt
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  • No -- the equation $ax+by+c=0$ represents one line somewhere in the plane. (Which line it is depends on what $a$, $b$ and $c$ are). – hmakholm left over Monica Apr 11 '16 at 17:06
  • the equation ax+by+c = 0 has two unknown constants once a relationship between a,b,c is known then we are left with one unknown constant and equation with one unknown constant always represents a family of lines. – Matt Apr 11 '16 at 17:09
  • x @Raghav: What do you mean by "has two unknown constants"? Is there a secret context to your question that you're too lazy to actually explain in the question and instead expect a reader to figure out telepathically? – hmakholm left over Monica Apr 11 '16 at 17:11
  • ax+by+c=0 can be written as a/cx+b/cy+1=0 or Ax+By+1=0 where A,B are two unknown constants – Matt Apr 11 '16 at 17:14
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    What if $c=0$? :-P. – YoTengoUnLCD Apr 11 '16 at 17:20
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    @HenningMakholm, I don't understand the confusion or the 'tude here. $ax +by + c = 0$ can be viewed as representing a family, just like $x^2 + C$ (as in $\int 2x , dx = x^2 + C$) represents a family. OP is asking for help in verifying that all members of the family $ax+by+c = 0$ pass through a fixed point if and only if $a$, $b$, and $c$ are linearly related to each other. I feel like I'm just repeating OP but I don't know how it can be made more clear. –  Apr 11 '16 at 17:20
  • if c=0 then the equation is ax+by = 0 which already has only two unknown constants – Matt Apr 11 '16 at 17:21
  • Do not forget that we must have $(a,b) \ne (0,0)$ in order for the equation $ax+by+c=0$ to represent a line. (If both $a$ and $b$ are zero, we no longer have an equation of first degree, and in that case the solution is either all points (entire plane) or no points (empty set).) – Jeppe Stig Nielsen Nov 07 '16 at 13:05

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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.