Let A ( 6 , 8 ) , B ( 10 cos α , − 10 sin α ) and C ( − 10 sin α , 10 cos α ) , be the vertices of a triangle. If L ( a , 9 ) and G ( h , k ) be its orthocenter and centroid respectively, then ( 5 a −

Question

Let A ( 6 , 8 ) , B ( 10 cos α , − 10 sin α ) and C ( − 10 sin α , 10 cos α ) , be the vertices of a triangle. If L ( a , 9 ) and G ( h , k ) be its orthocenter and centroid respectively, then ( 5 a − 3 h + 6 k + 100 sin 2 α ) is equal to ______ -.

TestHub · Accepted Answer

​ 3 a + 0 ​ = h ⇒ a = 3 h 3 9 + 0 ​ = k ⇒ k = 3 ∵ ( h , k ) = ( 3 6 + 10 cos α − 10 sin α ​ , 3 8 − 10 sin α − 10 cos α ​ ) 6 + 10 cos α − 10 sin α = 3 h 10 cos α − 10 sin α = 3 h − 6 10 ( cos α − sin α ) = 1 3 8 − 10 sin α + 10 cos α ​ = k ⇒ 100 sin 2 α = 99 h = 3 7 ​ ⇒ a = 7 ​ Now, 5 a − 3 h + 6 k + 100 sin 2 α = 35 − 7 + 18 + 99 = 145

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