    Question

# If A(x1,y1),Bx2,y2),C(x3,y3) are the vertices of an equilateral triangle and (x1−2)2+(y1−3)2=(x2−2)2+(y2−3)2=(x3−2)2+(y3−3)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 3Here, △ABC is equilateral (x1−2)2+(y1−3)2=(x2−2)2+(y2−3)2=(x3−2)2+(y3−3)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  Suggest Corrections  1      Similar questions  Related Videos   MATHEMATICS
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