Question

If the orthocentre and centroid of a triangle are $(-3,5,1)$ and $(3,3,-1)$ respectively, then its circumcentre is

• A. $(6,2,-1)$
• B. $(1,2,0)$
• C. $(6,2,-2)$
• D. $(6,-2,2)$

Answer 1

The correct option is C $(6,2,-2)$

given $centroid\left(G\right)=(3,3,-1)$

$orthocenter\left(H\right)=(-3,5,1)$

let $dircumcenter\left(S\right)=(x,y,z)$

we know that G divides H,S in the ratio $m:n=2:1$

therefore $(3,3,-1)=\left(\right(\frac{mx2+nx}{m+n}),(\frac{my2+ny1}{m+n}),(\frac{mz2+nz1}{m+n}\left)\right)$

$(3,3,-1)=\left(\right(\frac{2x-3}{3}),(\frac{2y+5}{3}),(\frac{2z+1}{3}\left)\right)$

$(x,y,z)=(6,2,-2)$