Three lines $y-z-1=0, x=0 ; x+z+1=0, y=0 ; x-z-1=0, y=0$ intersect the $x y$ plane at $\mathrm{A}, \

Question

Three lines $y-z-1=0, x=0 ; x+z+1=0, y=0 ; x-z-1=0, y=0$ intersect the $x y$ plane at $\mathrm{A}, \mathrm{B}$ and C . If the orthocentre of $	riangle A B C$ is ( $\mathrm{p}, \mathrm{q}, \mathrm{r}$ ) then $3 p+q+r=$ $\_\_\_\_$

Professor PC Thomas Classes · Answer

Intersection of line $y-z-1=0, x=0$ with xy plane gives $x=0, y=1$ and $z=0$ so point $A$ is $(0,1,0)$. Similarly point $B$ is $(-1,0,0)$ and point $C$ is $(1,0,0)$. Here $A B^2+A C^2=B C^2$. So triangle ABC is right angled at $A$. Hence orthocentre is point $A(0,1,0)$.