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Question

If the centroid and circumcentre of a triangle are (3,3) and (6,2), respectively, then the orthocentre is

A
(3,5)
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B
(3,1)
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C
(3,1)
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D
(9,5)
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Solution

The correct option is A (3,5)
In any triangle, orthocentre, centroid and circumcentre are collinear and centroid divides the line joining orthocentre and circumcenter in the ratio 2:1.
Let the orthocentre be (x,y).
Using the section formula, (mx2+nx1m+n,my2+ny1m+n)

Substituting (x1,y1)=(x,y) and (x2,y2)=(6,2) and m=2,n=1 in the section formula, we get the centroid =(2(6)+1(x)2+1,2(2)+1(y)2+1)=(x+123,y+43)

Given centroid =(3,3)

=>(x+123,y+43)=(3,3)
=>x+123=3;y+43=3
=>x+12=9;y+4=9
=>x=3;y=5
Hence, orthocentre =(3,5)

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