Question
If O is the origin and OP, OQ are distinct tangents to the circle x^{2} + y^{2} + 2gx + 2fy + c = 0, the circumcentre of the triangle OPQ is

(–g, –f)

(g, f)

(–f, –g)

None of these.
medium
Solution
None of these.
Since PQ is the chord of contact of the tangents from the origin O to the circle
x^{2} + y^{2} + 2gx + 2fy + c = 0, (1)
equation of PQ is
gx + fy + c = 0 (2)
An equation of a circle through the intersection of (1) and (2) is given by
If the circle (3) passes through O, the origin, then c + λc = 0, i.e.,
λ = –1, and the equation of the circle (3)
becomes x^{2} + y^{2} + gx + fy = 0
Centre of the circle is (–g/2, –f/2), and hence it is the circumcentre of the triangle OPQ.
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