Question bank/JEE Main/Mathematics

Let H be the orthocenter of $\mathrm{\Delta}ABC$ where
$A \equiv (0,0),B \equiv (8,0)$ and $C \equiv (4,4\sqrt{3})$. If
$G_{1},G_{2},G_{3},\ldots\ldots\ldots\$ are centroids of
$\mathrm{\Delta}AHB,\ \ \mathrm{\Delta}AG_{1}B,\ \ \Delta AG_{2}B,\ldots\ldots\ldots\ldots$
respectively then find value of
${Lim}_{n \rightarrow \infty}{\sum_{i = 1}^{n}\mspace{2mu}\ Ar}\mathrm{\Delta}AG_{i}B$

ekuk H, $\mathrm{\Delta}ABC$ dk yEcdsUnz gS] tgk¡
$\mathrm{\Delta}AHB,\ \ \mathrm{\Delta}AG_{1}B,\ \ \Delta AG_{2}B,\ldots\ldots\ldots\ldots$
Unzd gS rc
${Lim}_{n \rightarrow \infty}{\sum_{i = 1}^{n}\mspace{2mu}\ Ar}\mathrm{\Delta}AG_{i}B$
dk eku Kkr dhft, &
[uksV $Ar(\Delta AG_{i}B)$ f=Hkqt $\Delta AG_{i}B$

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$\mathrm{\Delta}ABC$ is equilateral $\Rightarrow$ H is centroid
of
$\bigtriangleup ABC \Rightarrow ar(\Delta AHB) = \frac{1}{3}ar(\Delta ABC) = \frac{16}{\sqrt{3}}$
Now $G_{1}$ is centroid of
$\Delta ABC \Rightarrow ar\left( \Delta AG_{1}B \right) = \frac{1}{3}\ ar(\Delta AHB) = \frac{1}{3}\left( \frac{16}{\sqrt{3}} \right)$

Source: Grass-G / M-25 Mathematics Paper 7 Jan.docx