Let medians L1, L2 and L3 of a triangle A B C belong to the family of lines 2 x-y+2+λ(x+2 y+ 1)=0 where λ is a parameter. Points P, Q, R are the mid-points of sides B C, C A and A B respectively where P ≡(2,6) and Q=(-5,2). If area of triangle B R G is 6 λ where G is centroid of triangle A B C, then find the value of λ.

Question

Let medians L1, L2 and L3 of a triangle A B C belong to the family of lines 2 x-y+2+λ(x+2 y+ 1)=0 where λ is a parameter. Points P, Q, R are the mid-points of sides B C, C A and A B respectively where P ≡(2,6) and Q=(-5,2). If area of triangle B R G is 6 λ where G is centroid of triangle A B C, then find the value of λ. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

2 x - y +2+λ( x +2 y +1)=0

operatornameAr(Δ BRG )= operatornameAr(Δ AQG )=(1/2)|-7 -12 1 -1 0 1 -5 2 1| ⇒ 30=6 λ ⇒ λ=5

2x−y+2+λ(x+2y+1)=0 Ar(ΔBRG)=Ar(ΔAQG)=21​∣∣​−7−1−5​−1202​111​∣∣​ ⇒30=6λ⇒λ=5

$2 x - y + 2 + λ (x + 2 y + 1) = 0$

$Ar (Δ BRG) = Ar (Δ A QG) = \frac{1}{2} ∣ ∣ - 7 - 1 - 5 - 12 02 111 ∣ ∣$
$\Rightarrow 30 = 6 λ \Rightarrow λ = 5$