Let the triangle PQR be the image of the triangle with vertices (1,3),(3,1) and (2,4) in t…

Question

Let the triangle PQR be the image of the triangle with vertices $(1,3),(3,1)$ and $(2,4)$ in the line $x+2 y=2$. If the centroid of $	riangle \mathrm{PQR}$ is the point $(\alpha, \beta)$, then $15(\alpha-\beta)$ is equal to :

Accepted Answer

Let ' $G$ ' be the centroid of $\Delta$ formed by $(1,3)(3,1)$ \& $(2,4)$

$$ \mathrm{G} \cong\left(2, \frac{8}{3}\right) $$

Image of G w.r.t. $x+2 y-2=0$

$$ \begin{aligned} & \frac{\alpha-2}{1}=\frac{\beta-\frac{8}{3}}{2}=-2 \frac{\left(2+\frac{16}{3}-2\right)}{1+4} \ & =\frac{-2}{5}\left(\frac{16}{3}\right) \ & \Rightarrow \alpha=\frac{-32}{15}+2=\frac{-2}{15}, \beta=\frac{-32 \times 2}{15}+\frac{8}{3}=\frac{-24}{15} \ & 15(\alpha-\beta)=-2+24=22 \end{aligned} $$

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