If origin is the centroid of the triangle \(P Q R\) with vertices \(P(2 a, 2,6), Q(-4,3 b,-10)\) and \(R(8…
If origin is the centroid of the triangle \(P Q R\) with vertices \(P(2 a, 2,6), Q(-4,3 b,-10)\) and \(R(8,14,2 c)\), then the values of \(a, b, c\) respectively are
\(2, \frac{16}{3},-2\)
\(-2,-\frac{16}{3},-2\)
\(-2,-\frac{16}{3}, 2\)
\(-2, \frac{16}{3},-2\)
Solution
Centroid is given by,
\(\begin{aligned}
& x=\frac{x_1+x_2+x_3}{3} \\
& y=\frac{y_1+y_2+y_3}{3} \\
& z=\frac{z_1+z_2+z_3}{3}
\end{aligned}\)
Using values given,
\(0=\frac{2 a-4+8}{3} \Rightarrow a=-2\)
Also, \(\quad 0=\frac{2+3 b+14}{3}, b=\frac{-16}{3}\)
and \(\quad 0=\frac{6-10+2 C}{3}, c=2\)