Question

The lines $x+y=0, \, x-4y=0$ and $2x-y=0$ are the altitudes of a triangle. If one of the vertices has the coordinates $\left(- \lambda , \lambda \right)$ and the locus of the centroid of this triangle is $ax+by=0$ (where $a$ and $b$ are positive integers and coprime to each other), then the value of $\left(a + 2 b\right)$ is

Answer 1

Equation of $AD,$ $x+y=0$

Equation of $B E, x-4 y=0$
Equation of $C F, 2 x-y=0$
Let the centroid is $\left(x_1, y_1\right)$
$ \begin{array}{l} \therefore x_1=\frac{-\lambda+4 \mu+\delta}{3} \\ \text { and } y_1=\frac{\mu+\lambda+2 \delta}{3} \ldots \text { (i) } \\ \because B C \perp A D ; \frac{\mu-2 \delta}{4 \mu-\delta}=1 \Rightarrow 3 \mu=-\delta \\ A C \perp B E ; \frac{2 \delta-\lambda}{\delta+\lambda}=-4 \Rightarrow \lambda=-2 \delta \\ A B \perp C F, \frac{\mu-\lambda}{4 \mu+\lambda}=-\frac{1}{2} \Rightarrow \lambda=6 \mu \end{array} $
Now using (i) and (ii) we get
$ \begin{array}{l} \frac{x_1}{y_1}=-5 \\ \Rightarrow x+5 y=0 \end{array} $