If $G(\bar{g}), H(\bar{h})$ and $P(\bar{p})$ are respectively centroid, orthocenter and circumcentre of a…
If $G(\bar{g}), H(\bar{h})$ and $P(\bar{p})$ are respectively centroid, orthocenter and circumcentre of a triangle and $x \bar{p}+y \bar{h}+z \bar{g}=\overline{0}$, then $x, y, z$ are respectively.
$1,1,-2$
$1,3,-4$
$2,1,-3$
$2,3,-5$
Solution
We know that centroid divides a line joining orthocenter to circumcentre in the ratio $2: 1$.
$\begin{aligned}
& \therefore \overline{\mathrm{g}}=\frac{\overline{\mathrm{h}}+2 \overline{\mathrm{p}}}{1+2} \Rightarrow 2 \overline{\mathrm{p}}+\overline{\mathrm{h}}=-3 \overline{\mathrm{g}} \\
& \therefore \mathrm{x}=2, \mathrm{y}=1, \mathrm{z}=-3 \text { as per data given. }
\end{aligned}$