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  • If is the origin and <img alt="\mathrm{O P, O Q}" src="https://entrancecorner.oncodecogs.com/gif.late

If \mathrm{O} is the origin and \mathrm{O P, O Q} are the tangents from the origin to the circle \mathrm{x^2+y^2-6 x+4 y+8=0,} then the circumcentre of the triangle \mathrm{O P Q} is
 

Option: 1

(3,-2)

 


Option: 2

\mathrm{\left(\frac{3}{2},-1\right)}
 


Option: 3

\mathrm{\left(\frac{3}{4},-\frac{1}{2}\right)}
 


Option: 4

\mathrm{\left(-\frac{3}{2}, 1\right)}


Answers (1)

best_answer

We note that \mathrm{P Q} is the chord of contact of the tangents from the origin to the circle

\mathrm{ x^2+y^2-6 x+4 y+8=0 }...........(i)

Equation of \mathrm{ P Q \: is\: 3 x-2 y-8=0 }.............(ii)

Equation of a circle passing through the intersection of (i) and (ii) is

\mathrm{ x^2+y^2-6 x+4 y+8+\lambda(3 x-2 y-8)=0}..........(iii)

If this represents the circumcircle of the triangle \mathrm{ O P Q}, it passes through \mathrm{O(0,0)}

and the equation (iii) becomes \mathrm{x^2+y^2-3 x+2 y=0}

Coordinates of the circumcentre are \mathrm{\left(\frac{3}{2},-1\right)}

Hence option 2 is correct.

Posted by

vishal kumar

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