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If A(x1,y1),Bx2,y2),C(x3,y3) are the vertices of an equilateral triangle and (x12)2+(y13)2=(x22)2+(y23)2=(x32)2+(y33)2, then

A
Coordinates of orthocentre is (0,0)
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B
Coordinates of circumcentre is (2,3)
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C
value of (x1+x2+x3)+3(y1+y2+y3) is 33
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D
value of 2(x1+x2+x3)(y1+y2+y3) is 3
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Solution

The correct options are
B Coordinates of circumcentre is (2,3)
C value of (x1+x2+x3)+3(y1+y2+y3) is 33
D value of 2(x1+x2+x3)(y1+y2+y3) is 3
Here, ABC is equilateral
(x12)2+(y13)2=(x22)2+(y23)2=(x32)2+(y33)2
A,B,C are equidistant from the point (2,3)
(2,3) is the circumcentre
Also (2,3) will be the coordinates of centroid and orthocentre.
( in equilateral triangle circumcentre, orthocentre and centroid coincides with each other )

Now
(x1+x2+x33,y1+y2+y33)=(2,3)x1+x2+x3=6,y1+y2+y3=9(x1+x2+x3)+3(y1+y2+y3)=332(x1+x2+x3)(y1+y2+y3)=3

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