Let n C r − 1 = 28 , n C r = 56 and n C r + 1 = 70 . Let A ( 4 cos t , 4 sin t ) , B ( 2 sin t , − 2 cos t ) and C ( 3 r − n , r 2 − n − 1 ) be the vertices of a triangle A BC , where t is a par

Question

Let n C r − 1 ​ = 28 , n C r ​ = 56 and n C r + 1 ​ = 70 . Let A ( 4 cos t , 4 sin t ) , B ( 2 sin t , − 2 cos t ) and C ( 3 r − n , r 2 − n − 1 ) be the vertices of a triangle A BC , where t is a parameter. If ( 3 x − 1 ) 2 + ( 3 y ) 2 = α , is the locus of the centroid of triangle ABC , then α equals

TestHub · Accepted Answer

n C r − 1 ​ = 28 n C r ​ = 56 n C r + 1 ​ = 70 n C r − 1 ​ n C r ​ ​ = 56 28 ​ ⇒ n − r + 1 r ​ = 2 1 ​ n C r + 1 ​ n C r ​ ​ = 70 56 ​ ⇒ n − r r + 1 ​ = 56 70 ​ ​ ⎭ ⎬ ⎫ ​ n = 8 = 3 h = 3 4 c o s t + 2 s i n t + 1 ​ k = 3 4 s i n t − 2 c o s t ​ 3 h − 1 = 4 cos t + 2 sin t 3 k − 1 = 4 sin t − 2 cos t ​ ( 1 ) 2 + ( 2 ) 2 ( 3 h − 1 ) 2 + ( 3 k ) 2 = 20 ​ Locus of centroid: ( 3 x − 1 ) 2 + ( 3 y ) 2 = 20 ⇒ α = 20

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