Question

The slopes of sides $BC , CA , AB$ of triangle $ABC$ whose orthocentre is origin are $-1,-2,-3$ respectively. If locus of centroid of triangle $A B C$ is $y=\left(\frac{m}{n}\right) x$, where $m, n$ are relatively prime then find $(n-m)$.

Answer 1

Equation of altitudes through A, B, C are
$y=x, x=2 y, x=3 y $
$\text { let } A (\alpha, \alpha), B (2 \beta, \beta), C (3 \gamma, \gamma) $
$\therefore \text { Slope of } AB =\frac{\beta-\alpha}{2 \beta-\alpha}=-3 \Rightarrow \beta=\frac{4 \alpha}{7} $
$\text { Slope of } B C=\frac{\gamma-\beta}{3 \gamma-2 \beta}=-1 \Rightarrow \frac{7 \gamma-4 \alpha}{21 \gamma-8 \alpha}=-1 \Rightarrow \gamma=\frac{3}{7} \alpha $
$\therefore \text { Centroid }=\left(\frac{\alpha+2 \beta+3 \gamma}{3}, \frac{\alpha+\beta+\gamma}{3}\right)=( x , y )$
$\therefore \frac{y}{x}=\frac{\alpha+\beta+\gamma}{\alpha+2 \beta+3 \gamma}=\frac{\alpha+\frac{4 \alpha}{7}+\frac{3 \alpha}{7}}{\alpha+\frac{8 \alpha}{7}+\frac{9 \alpha}{7}}=\frac{14 \alpha}{24 \alpha}=\frac{7}{12}=\frac{m}{n}$
$\therefore ( n - m )=12-7=5$