Question

If $G\left(\overline{g}\right),H\left(\overline{h}\right),P\left(\overline{p}\right)$ are centroid, orthocenter and circumcenter of a triangle and $x\overline{p}+y\overline{h}+z\overline{g}=0$ then $(x,y,z)=$..................

• A. $1,1,-2$
• B. $2,1,-3$
• C. $1,3,-4$
• D. $2,3,-5$

Answer 1

Answer: (B)

$2,1,-3$

$G\left(\overline{g}\right),H\left(\overline{h}\right),P\left(\overline{p}\right)$ are centroid, orthocenter and circumcenter of a triangle.

Also, $x\overline{p}+y\overline{h}+z\overline{g}=0$ .... $\left(i\right)$

We know that, for any triangle

$HG=2GP$

$⟹\overline{g}-\overline{h}=2(\overline{p}-\overline{g})$

$⟹\overline{g}-\overline{h}=2\overline{p}-2\overline{g}$

$⟹2\overline{p}-2\overline{g}-\overline{g}+\overline{h}=0$

$⟹2\overline{p}+\overline{h}-3\overline{g}=0$

Comparing above equation with $\left(i\right)$, we get

$x=2,y=1$ and $z=-3$

Hence, $(x,y,z)=(2,1,-3)$