$\text{ The vertices of a }\Delta ABC\text{ are }A(1,0,0),B(0,2,0),C(0,0,3)\text{ . If }a,b,-111\text{ are the direction }$$\text{ ratios of the line joining the orthocenter and circumcentre of }\Delta ABC\text{ then }a+b=$

$\text{ The equation of plane }\mathrm{ABC}\text{ is }\frac{x}{1}+\frac{y}{2}+\frac{z}{3}=1\text{ if }H(\alpha ,\beta ,\gamma )\text{ is the orthocenter then }$

$AH\perp BC,BH\perp CA\Rightarrow \alpha =2\beta =3\gamma$

$\mathrm{H}\text{ lies in the plane of triangle }\mathrm{ABC}\text{ then }\frac{\alpha }{1}+\frac{\alpha }{4}+\frac{\alpha }{9}=1$

$$\begin{gathered} \Rightarrow \alpha =\frac{36}{49} \\ \therefore H=\left(\frac{36}{49},\frac{18}{49},\frac{12}{49}\right) \end{gathered}$$

$\text{ The circum center, orthocenter, centroid are collinear and centroid }\left(G\right)=\left(\frac{1}{3},\frac{2}{3},1\right)$

$\text{ The dr'sof }HG,\frac{36}{49}-\frac{1}{3},\frac{18}{49}-\frac{2}{3},\frac{12}{49}-1=59,-44,-111$

$a=59,b=-44\Rightarrow a+b=15$