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Question

If O is the origin and OP, OQ are the tangents from the origin to the circle x2+y26x+4y+8=0, the circumcenter of the triangle OPQ is

A
(3,2)
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B
(32,1)
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C
(34,12)
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D
(32,1)
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Solution

The correct option is B (32,1)
Now PQ is the chord of contact of the tangents from the origin to the circle x2+y26x+4y+8=0(1)
Equation of PQ is 3x2y8=0(2)
Equation of a circle passing through the intersection of (1) and (2) is
x2+y26x+4y+8+λ(3x2y8)=0(3)
if this represents the circumcircle of the triangle OAB, it passes through O(0, 0), so λ=1 and the equation (3) becomes x2+y23x+2y=0
The required coordinates of the centre are (32,1)

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