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  • The equations of three circles are <img alt="\mathrm{ x^2+y^2-12 x-16 y+64=0, \quad 3 x^2+3 y^2-36 x+81=0 }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20x%5E2&plus;y%5E2

The equations of three circles are \mathrm{ x^2+y^2-12 x-16 y+64=0, \quad 3 x^2+3 y^2-36 x+81=0 } and

\mathrm{x^2+y^2-16 x+81=0}. The coordinates of the point from which the length of tangents drawn to each of the three circles is equal

Option: 1

\left(\frac{33}{4}, 2\right)


Option: 2

(2,2) \quad


Option: 3

\left(2, \frac{33}{4}\right)


Option: 4

None of these


Answers (1)

best_answer

The required point is the radical centre of the three given circles

Now, \mathrm {S_1-S_2=0 \Rightarrow-16 y+37=0}                          \mathrm{S_2-S_3=0 \Rightarrow 4 x-54=0} and

  \mathrm{S_3-S_1=0 \Rightarrow-4 x+16 y+17=0}

Solving these equations, we get \mathrm{x=\frac{54}{4}, y=\frac{37}{16} \Rightarrow x=\frac{27}{2}, y=\frac{37}{16}} .

Hence the required point is \mathrm{\left(\frac{27}{2}, \frac{37}{16}\right)}

Posted by

Gautam harsolia

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