Question

A line with positive rational slope passes through the point $A(6,0)$ and is at a distance of $5$ units from $B(1,3)$. If the slope of the line is $m$, then the value of $75m$ is ______.

Answer 1

Answer: 40

Equation of line passing through point $A(6,0)$ and having slope $m$ is
$(y-0)=m(x-6),m>0$
or $mx-y-6m=0,m>0$ ..... (1)
Distance of this line from point $B(1,3)$ is $5$ .
So,$\left|\frac{m-3-6m}{\sqrt{1+m^2}}\right|=5$
$\Rightarrow (3+5m{)}^2=25\left(1+m^2\right)$
$\Rightarrow 9+30m=25$
$\Rightarrow m=\frac{16}{30}=\frac{8}{15}$