If ( α , β ), ( ¯ x , ¯ y ) a n d ( u , v ) are respectively coordinates of the circumcentre, centroid and orthocentre of a triangle. Options: A . &nbsp3¯x =2α+u and 3¯y =2β+v B . &nbsp3¯x =2α−u and 3¯y =2β−v C . &nbsp3¯x =2α−u and 3¯y =2β+v D . &nbsp3¯x =2α+u and 3¯y =2β−v

If (α, β), (¯x , ¯y) and (u

Question

If (

α, β

(¯ x, ¯ y) a n d (u, v)

are respectively coordinates of the circumcentre, centroid and orthocentre of a triangle.

Options:

A . 3¯x =2α+u and 3¯y =2β+v

B . 3¯x =2α−u and 3¯y =2β−v

C . 3¯x =2α−u and 3¯y =2β+v

D . 3¯x =2α+u and 3¯y =2β−v

Answer: Option C
:
C
We know that, the centroid of a triangle divides the segment joining the orthocentre and circumcentre internally in the ratio 2 : 1. Therefore,

¯ x = \frac{2 α + u}{2 + 1} a n d ¯ y = \frac{2 β + v}{2 + 1}

\Rightarrow 3 ¯ x = 2 α + u a n d ¯ y = 2 β + v

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