The locus of the centroid of the triangle whose vertices are (3 cos α , 3 sin α ) , (9 sin α , - 9 cos α ) and (1 , 0) is a circle of radius R , then the value of (9 R2/2) is equal to.

Question

The locus of the centroid of the triangle whose vertices are (3 cos α , 3 sin α ) , (9 sin α , - 9 cos α ) and (1 , 0) is a circle of radius R , then the value of (9 R2/2) is equal to. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-tkzvyrot9euljmdv.jpg" /> Let centroid of Δ ABC is (h , k) ⇒ h=(3 cos α + 9 sin α + 1/3) & k=(3 sin α - 9 cos α + 0/3) ⇒ 3(cos α + 3 sin α )=3h-1 & 3(sin α - 3 cos α )=3k ⇒ 9(cos α + 3 sin α )2=(3 h - 1)2 & 9(sin α - 3 cos α )2=9k2 ⇒ (3 h - 1)2+9k2=9[(cos α + 3 sin α )2 + (sin α - 3 cos α )2] Replace h by x and k by y ⇒(3 x-1)2+9 y2=90( cos 2 α+ sin 2 α) ⇒ (x - (1/3))2+y2=(90/9) ⇒ (x - (1/3))2+y2=10 ⇒ R=√10 ⇒ (9 R2/2)=(9 (10)/2)=45

Q. The locus of the centroid of the triangle whose vertices are (3cosα,3sinα) , (9sinα,−9cosα) and (1,0) is a circle of radius R , then the value of 29R2​ is equal to.

Q. The locus of the centroid of the triangle whose vertices are $(3 cos α, 3 s in α)$ , $(9 s in α, - 9 cos α)$ and $(1, 0)$ is a circle of radius $R$ , then the value of $\frac{9 R ^{2}}{2}$ is equal to.