<s>[INST] Find the minimum length of a line segment with endpoints on the coordinate axes that passes through the point $(1, 1)$ A line passes through the point $P=(1,1)$ and through the $x$ and $y$-axes at points $A$ and $B$ respectively. Find the minimum length of the line segment $AB$.
 [/INST] $\frac{y-1}{x-1}=m$ defines the line.
$A$ is the $x$ intercept which is $(1-\frac{1}{m}, 0)$ and $B$ is the $y$ intercept which is $(0,1-m)$.
By minimizing the distance between these two points we can calculate $m$ which will then give us the distance. To make the algebra simple we will actually minimize the distance squared, call it d.
Then $d=(\frac{1}{m}-1)^2+(1-m)^2$ and taking the derivative with respect to $m$ gives $\frac{dd}{dm}=-\frac{2}{m^2}(\frac{1}{m}-1)-2(1-m)=\frac{-2+2m}{m^3}-\frac{2m^3(1-m)}{m^3}=\frac{-2+2m-2m^3+2m^4}{m^3}$
Setting this equal to $0$ gives $-2+2m-2m^3+2m^4=0\implies -1+m-m^3+m^4\implies m = 1$ or $m=-1$.
If $m=1$ then both the $x$ and $y$ intercepts will be $0$ so $A=B$ and the distance is $0$.
If $m=-1$ then $A=(2,0)$ and $B=(0, 2)$ so the distance is $\sqrt{2^2+2^2}=\sqrt{8}=2\sqrt{2}$
Therefore, the minimum distance is $0$ when $A=B$. If we change the problem to say that $A$ and $B$ are distinct then the minimum distance is $2\sqrt{2}$.
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