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Question

If coordinates of the vertices of a triangle are (2,0),(6,0) and (1,5), then distance between its orthocentre and circumcentre is

A
4
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B
6
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C
5
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D
None of these
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Solution

The correct option is A 4
Let the vertices of triangle be A(2,0),B(6,0),C(1,5)
Let the cir-cum center be P(h,k)
Then,
AP=BP
AP2=BP2
(h2)2+k2=(h6)2+k2
h=4
Also, AP=CP
AP2=CP2
(h2)2+k2=(h1)2+(k5)2
h2+44h+k2=h2+12h+k2+2510k
k=3
So, cir-cum center of the triangle ABC is (4,3)
Centroid (G) of the ΔABC=(2+6+13,0+0+53)=(3,53)
We know that,
Distance between the centroid and orthocenter is equal to the twice of the distance between the centroid and the circumcentre
So, Distance between the orthocentre and circumcentre =3×GC=3×(43)2+(353)2=5units

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