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 $\left(a+2b\right)$ 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 $\left(x_1,y_1\right)$
$\begin{array}{l}\therefore x_1=\frac{-\lambda +4\mu +\delta }{3}\\ \text{ and }{y}_1=\frac{\mu +\lambda +2\delta }{3}\dots \text{ (i) }\\ \because BC\perp AD;\frac{\mu -2\delta }{4\mu -\delta }=1\Rightarrow 3\mu =-\delta \\ AC\perp BE;\frac{2\delta -\lambda }{\delta +\lambda }=-4\Rightarrow \lambda =-2\delta \\ AB\perp CF,\frac{\mu -\lambda }{4\mu +\lambda }=-\frac{1}{2}\Rightarrow \lambda =6\mu \end{array}$
Now using (i) and (ii) we get
$\begin{array}{l}\frac{x_1}{y_1}=-5\\ \Rightarrow x+5y=0\end{array}$