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 $\left(a+2b\right)$ is

Answer 1

Answer: 11

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)
$\because BC\perp AD;$ $\frac{\mu -2\delta }{4\mu -\delta }=1\Rightarrow 3\mu =-\delta$
$AC\perp BE;$ $\frac{2\delta -\lambda }{\delta +\lambda }=-4\Rightarrow \lambda =-2\delta$
$AB\perp CF;$ $\frac{\mu -\lambda }{4\mu +\lambda }=-\frac{1}{2}\Rightarrow \lambda =6\mu$
Now using (i) and (ii) we get
$\frac{x_1}{y_1}=-5$
$\Rightarrow \boxed{x+5y=0}$