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=((m/n)) x, where m, n are relatively prime then find (n-m).

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=((m/n)) x, where m, n are relatively prime then find (n-m). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Equation of altitudes through A, B, C are

y = x, x = 2 y, x = 3 y

let A (α, α), B (2 β, β), C (3 γ, γ)

∴ Slope of A B = \frac{β - α}{2 β - α} = - 3 \Rightarrow β = \frac{4 α}{7}

Slope of BC = \frac{γ - β}{3 γ - 2 β} = - 1 \Rightarrow \frac{7 γ - 4 α}{21 γ - 8 α} = - 1 \Rightarrow γ = \frac{3}{7} α

∴ Centroid = (\frac{α + 2 β + 3 γ}{3}, \frac{α + β + γ}{3}) = (x, y)

∴ \frac{y}{x} = \frac{α + β + γ}{α + 2 β + 3 γ} = \frac{α + \frac{4 α}{7} + \frac{3 α}{7}}{α + \frac{8 α}{7} + \frac{9 α}{7}} = \frac{14 α}{24 α} = \frac{7}{12} = \frac{m}{n}

∴ (n - m) = 12 - 7 = 5

Q. 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 ABC is y=(nm​)x, where m,n are relatively prime then find (n−m).

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