General Form of the equation of a straight line.

Question

We all know the general form of the equation of a straight line which is

$A x + B y + C = 0$

but my question is, what $A$ represent and what $B$ represent and what $C$ represent.

Sorry I am not good in mathematics so I need your help.

Clear example with images would be appreciated.

Thank you.

Answer 1

$A,B$ and $C$ do not represent anything, if you consider each one of them separately. Note that you can multiply the equation $Ax+By+C=0$ for any nonzero number $k$ to get $kAx+kBy+kC=0,$ which has the same solutions and, so, define the same line. What have a geometric interpretation are the quotients $-\frac{A}{B}$ and $-\frac{C}{B},$ assuming $B\ne 0.$ In such a case you have $y=-\frac{A}{B}x-\frac{C}{B}.$ Then $-\frac{A}{B}$ is the slope of the line and $-\frac{C}{B}$ is the $y$-coordinate of the point of intersection of the line with the $y$-axis. (That is, $\left(0,-\frac{B}{A}\right)$ is a point of the line.)

In case $B=0$ the line has equation $Ax+C=0,$ or $x=-\frac{C}{A}$ and it is a vertical line that intersects the $x$-axis at the point $\left(-\frac{C}{A},0\right).$

Answer 2

Think of $c$ as an initial point on the line so let $c = (x_0, y_0)$. The $a$ and $b$ value tells value tells you how much to move in the $x$ direction and $y$ direction so that you will reach another point on the line. The overall vector equation of the line is $r=r_0+tv$ where $r_0$ is the initial point which is your $c$ and vector $v=(x, y, z,..., n)$. The $t$ represents your $A$ and $B$ value which tells you "how much to move in the directions $(x, y, z,..., n)$.

Answer 3

As @mfl pointed out, to get meaning you need to combine $A$, $B$, and $C$. Here is another meaning for them if you rescale them all by dividing both sides of the equation by $\sqrt {A^2+B^2}$.

${A \over \sqrt {A^2+B^2}}$ is the cosine of the direction angle of the vector perpendicular to the given line, while ${B \over \sqrt {A^2+B^2}}$ is the sine of the same angle.

${C \over \sqrt {A^2+B^2}}$ is the distance between the origin and the line (or its negative).

This may seem weird but I have found this interpretation useful in problems.