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If O is the origin and OP, OQ are distinct tangents to the circle \mathrm{x^2+y^2+2 g x+2 f y+c=0, } then the circumcentre of the triangle OPQ is

Option: 1

\mathrm{(-g,-f)}


Option: 2

\mathrm{(g, f)}


Option: 3

\mathrm{(-f,-g)}


Option: 4

\mathrm{\text { None of these }}


Answers (1)

best_answer

Since PQ is the chord of contact of the tangents from the origin O to the circle

\mathrm{ x^2+y^2+2 g x+2 f y+c=0 }                       ...(i)

\therefore  Equation of PQ is \mathrm{g x+f y+c=0}             ...(ii)

An equation of a circle through the intersection of (i) and (ii) is given by

\mathrm{ x^2+y^2+2 g x+2 f y+c+\lambda(g x+f y+c)=0 }        ...(iii)

If the circle (iii) passes through O, the origin, then \mathrm{c+\lambda c=0}, i.e., \mathrm{\lambda=-1, }and the equation of the circle (iii)

  1. becomes \mathrm{x^2+y^2+g x+f y=0}

Centre of this circle is \mathrm{(-g / 2,-f / 2)} and hence it is the circumcentre of the triangle OPQ.

Posted by

vishal kumar

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