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Q.

The orthocentre and the centroid of $Δ A B C$ are (5,8) $(3, \frac{14}{3})$ respectively. The equation of the side BC is $x - y = 0$ . Given that the image of the orthocentre of a triangle with respect to any side lies on the circumcircle of that triangle, then the diameter of the circumcircle of $Δ A B C$ is

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a

$8 \sqrt{10}$

b

$4 \sqrt{10}$

c

$\sqrt{10}$

d

$2 \sqrt{10}$

answer is C.

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Detailed Solution

$O (5, 8) G (3, \frac{14}{3})$

$S (α, β)$ is circumcentre

G is divides OS in the ratio 2:1

$S (α, β) = (2, 3)$

Image O (5,8) w.r.t line $y = x$ is A (8,5)

$∴$ Point A is lies on circle

radius of circle $r = A S = \sqrt{40}$

Diameter of circle $2 r = 4 \sqrt{10}$

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The orthocentre and the centroid of ΔABC are (5,8) 3,143 respectively. The equation of the side BC is x-y=0. Given that the image of the orthocentre of a triangle with respect to any side lies on the circumcircle of that triangle, then the diameter of the circumcircle of ΔABC is