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The centre of the circle, which cuts orthogonally each of the three circles

$\mathrm{x^2+y^2+2 x+17 y+4=0, \quad x^2+y^2+7 x+6 y+11=0}$ and $\mathrm{x^2+y^2-x+22 y+3=0}$ is
Option: 1
$(3,2)$

Option: 2
$(1,2)$

Option: 3
$(2,3)$

Option: 4
$(0,2)$

Answers (1)

Let the circle is $\mathrm{x^2+y^2+2 g x+2 f y+c=0}$ $\mathrm{.....(i)}$

Circle (i) cuts orthogonally each of the given three circles. Then according to condition

$\mathrm{2 g_1 g_2+2 f_1 f_2=c_1+c_2}$

$\mathrm{2 g+17 f=c+4}$ $\mathrm{.....(ii)}$

$\mathrm{7 g+6 f=c+11}$ $\mathrm{.....(iii)}$

$\mathrm{-g+22 f=c+3}$ $\mathrm{.....(iv)}$

On solving (ii), (iii) and (iv), $\mathrm{g=-3, f=-2}$ . Therefore, the centre of the circle $\mathrm{(-g,-f)=(3,2)}$

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