Six points (x i , y i ), i = 1, 2,..., 6 are taken on the circle x 2 + y 2 = 4 such that 
and 
. The line segment joining orthocentre of a triangle made by any three points and the centroid of the triangle made by other three points passes through a fixed points (h, k), then h + k is_________
Text Solution
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Sol. Let 
and 
.
Let O be the orthocentre of the triangle made by (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 )
⇒ O is (x 1 + x 2 + x 3 , y 1 + y 2 + y 3 ) ≡ ( α 1 , β 1 )
Similarly let G be the centroid of the triangle made by other three points
⇒ G is 
⇒ G is 
.
The point dividing OG in the ratio 3 : 1 is 
≡ (2, 1)
⇒ h + k = 3
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