# A vertical line in a cartesian coordinate system

Let's say I have points $A(a,a)$ and $B(a,0)$. What is the equation of the line $AB$? If I'm correct the slope is infinite, but it never has a y-intercept. This would give $y=\infty x$, but there are infinitely many lines which should have this formula but are different (they have a different $y$). Is this a flaw in the $y=mx+b$ system? How do we handle this?

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Thanks, I thought it'd be \inf :P – JohnPhteven Oct 23 '12 at 13:57

## 2 Answers

The equation of the line $AB$ is $x=a$. $x=0$ corresponds to the $y$-axis itself. We do not speak of the slope of a vertical line, so $y=mx+b$ does not apply here.

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Let the equation of the line be $Ax+By+C=0$

As it passes through $(a,a),(a,o)$ so, $Aa+Ba+C=0$ and $Aa+C=0\implies C=-Aa$

$Aa+Ba+C=0$ becomes $Ba=0\implies B=0$ as $a\ne 0$

So, the equation becomes $Ax+0y-Aa=0\implies x=a$ as $A=0$ would imply $C=0$.

Any line with $x$ co-ordinate same, can not be expressed as $y=mx+b$ as $\frac 1m$ becomes $\frac{a-a}{a-0}=\frac 0 a=0$ as $a\ne 0$

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I personally don't think it is appropriate to write $a/0=\infty$. Maybe you can say that the slope is not defined. Also, $(a-a)/(a-0)=0/a=0$ so you probably meant something else. – Pedro Tamaroff Oct 23 '12 at 22:56
@PeterTamaroff, what is the gradient of Y-axis? – lab bhattacharjee Oct 24 '12 at 4:25