Question

If the points $(2a,a),(a,2a)$ and $(a,a)$ enclose a triangle of area $18$ square units, then the centroid of the triangle is equal to

• A. (4, 4)
• B. (8, 8)
• C. (-4, -4)
• D. $(4\sqrt{2},4\sqrt{2})$
• E. (6, 6)

Answer 1

Answer: (B)

Given, that Area of triangle $=18$
$\Rightarrow \frac{1}{2}\left|\begin{array}{ccc}2a& a& 1\\ a& 2a& 1\\ a& a& 1\end{array}\right|=\pm 18$
$\Rightarrow \left|\begin{array}{ccc}2a& a& 1\\ a& 2a& 1\\ a& a& 1\end{array}\right|=\pm 36$
$\Rightarrow 2a(2a-a)-a(a-a)+1\left(a^2-2a^2\right)=\pm 36$
$\Rightarrow 2a^2-a^2=\pm 36$
$\Rightarrow a^2=\pm 36$
$\Rightarrow a^2=36$
$\Rightarrow a=\pm 6$
Now, centroid of the given triangle will be
$=\left(\frac{2a+a+a}{3},\frac{a+2a+a}{3}\right)=\left(\frac{4a}{3},\frac{4a}{3}\right)$
When $a=6$, centroid $=\left(\frac{4\times 6}{3},\frac{4\times 6}{3}\right)=(8,8)$