If $a,b$are intercepts of the line which is perpendicular to the line passing through the points $\left(3,5\right),\left(4,3\right)$ and passing through the point $\left(5,6\right)$then $\frac{a}{b}=$

$\text{ The slope of the line joining the points }(3,5),(4,3)\text{ is }m=\frac{3-5}{4-3}=-2$

Equation of line passing through the point $\left(5,6\right)$ and perpendicular to the line having slope $-2$  is  

$\begin{array}{c}y-6=\frac{1}{2}\left(x-5\right)\\ 2y-12=x-5\\ x-2y+7=0\end{array}$

Isolate the constant term and then divide with that constant on both sides 

$\begin{array}{c}x-2y=-7\\ \frac{x}{-7}+\frac{y}{\frac{7}{2}}=1\end{array}$

$\text{ Therefore, }x\text{-intercept is }a=-7\text{ and the }y\text{-intercept is }b=\frac{7}{2}$

Hence, 

$\frac{a}{b}=\boxed{-2}$