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  • Tangents drawn from the point P(1,8) to the circle <img alt="\mathrm{x^2+y^2-6 x-4 y-11=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%5E2&plus;y%5E2-6%20x-4%20y-11

Tangents drawn from the point P(1,8) to the circle \mathrm{x^2+y^2-6 x-4 y-11=0} touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is

Option: 1

x^2+y^2+4 x-6 y+19=0


Option: 2

x^2+y^2-4 x-10 y+19=0


Option: 3

x^2+y^2-2 x+6 y-29=0


Option: 4

x^2+y^2-6 x-4 y+19=0


Answers (1)

best_answer

Equation of AB the chord of contact is

\mathrm{ \begin{aligned} & 1 \cdot x+8 y-3(x+1)-2(y+8)-11=0 \\ & \Rightarrow-2 x+6 y-30=0 \Rightarrow x-3 y+15=0 \end{aligned} }

Equation of any circle through AB is

\mathrm{ x^2+y^2-6 x-4 y-11+\lambda(x-3 y+15)=0 \, \, It } will pass through P(1,8) if

\mathrm{ \begin{aligned} & 1+64-6-32-11+\lambda(1-24+15)=0 \\ & \Rightarrow 16-8 \lambda=0 \Rightarrow \lambda=2 \end{aligned} }

Thus, equation of the required circle is

 \mathrm{ \begin{aligned} & x^2+y^2-6 x-4 y-11+2(x-3 y+15)=0 \\ \Rightarrow & x^2+y^2-4 x-10 y+19=0 \end{aligned} }

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SANGALDEEP SINGH

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