Question

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

Answer 1

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

Equation of $BE, \, x-4y=0$
Equation of $CF, \, 2x-y=0$
Let the centroid is $\left(x_{1} , \, y_{1}\right)$
$\therefore x_{1}=\frac{- \lambda + 4 \mu + \delta}{3}$ ..... (i)
and $y_{1}=\frac{\mu + \lambda + 2 \delta}{3}$ ..... (ii)
$\because BC\bot AD;$ $\frac{\mu - 2 \delta}{4 \mu - \delta}=1\Rightarrow 3\mu =-\delta$
$AC\bot BE;$ $\frac{2 \delta - \lambda }{\delta + \lambda }=-4\Rightarrow \lambda =-2\delta$
$AB\bot CF;$ $\frac{\mu - \lambda }{4 \mu + \lambda }=-\frac{1}{2}\Rightarrow \lambda =6\mu $
Now using (i) and (ii) we get
$\frac{x_{1}}{y_{1}}=-5$
$\Rightarrow \boxed{x + 5 y = 0}$