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Let suppose that $R(x_1, y_1)$ is a point on the (x, y)plane and a line $L$ with slope $m$ passes through this point. There is a point $S(x_1, y_1)$ on $L$ such that $R$ and $S$ are coincident points. Then the point-slope equation can be given as:

$y_1-y_1 = m(x_1-x_1)$

If we don't know the value of $m$ then we do as follows:

$m = \frac {y_1-y_1} {x_1-x_1}$

$m = \frac 00$

But the division by $0$ is undefined. So I think it implies that for a point slope equation two points must not be coincident or their $x$ and $y$ coordinates must not be the same. Well, it is only my thinking, is there any condition like this in reality for a point slope equation?

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Samama Fahim
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  • To avoid such problems always define a line using the homogeneous coordinates $a$,$b$ and $c$ such that $a x+b y + c = 0$. The constraint to pass through R is then given by $c=-a x_1-b y_1$. – John Alexiou Apr 29 '13 at 14:22

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If $R$ and $S$ are coincident, the line through them is not defined. A line can have any slope and go though that point. This is what causes your division by zero.

You can certainly form the point-slope equation with just one point if you are given the slope. If you are not given the slope it takes a second point to determine it.

Ross Millikan
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    @SamamaFahim: Yes, there is. To determine a line you need two different points, not just two names for the same point. – Ross Millikan Apr 29 '13 at 14:21
  • So $y_1 - y_1 = m(x_1 - x_1)$ is valid if we're given the value of $m$? – Samama Fahim Apr 29 '13 at 14:54
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    @SamamaFahim: As you have written it, it is true regardless of the value of $m$. It says $0=m\cdot 0$. But it is not the equation of a line. – Ross Millikan Apr 29 '13 at 15:14
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    Yes, there is no need to recalculate slope if you are given slope. If given slope and one point $(x_0, y_0)$, you can use the point slope form of an equation $y - y_0 = m(x - x_0).$ If you don't know the slope, or it is not given, then you need two points to determine the line: in order to determine slope: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Then you can use this $m$, and one of the given points, to create the equation of the line: $y - y_1 = m(x - x_1)$, e.g. – amWhy Apr 29 '13 at 21:11