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(a) Find the equations of the two circles each of which touches both coordinate axes and passes through the point $(9,2)$.

(b) Find the coordinates of the second point of intersection of the two circles.

(c) Find an equation of the common chord in these two circles

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  • $\begingroup$ Have you done any work on this problem so far? $\endgroup$ – Newb Nov 3 '13 at 15:03
  • $\begingroup$ Yes, I have tried to solve the simultaneous equations (9-a)^2+(2-b)^2=r^2, (n_x-a)^2+b^2=r^2 and a^2+(n_y-b)^2+r^2, where n_x and n_y are the respective x and y coordinates of where the circle touches the axes, but I have always ended up with very messy expressions, instead of a numerical solution, as requested. $\endgroup$ – Giuseppe Nov 3 '13 at 15:05
  • $\begingroup$ Hint: When circles "touch" a line (say the axis), it does not just mean it intersects, it means the line (or axis) is also a tangent there. So equation of a circle which touches both axes is $(x-a)^2+(y \pm a)^2 = a^2$ (why?) $\endgroup$ – Macavity Nov 3 '13 at 15:33
  • $\begingroup$ (x-r)^2+(y-r)^2=r^2 therefore x^2+y^2+2xr+2yr+r^2=0 therefore substituting x=9 y=2, solving for r with the quadratic formula we obtain r=17, therefore equation of circle is (x-17)^2+(y-17)^2=17^2 $\endgroup$ – Giuseppe Nov 3 '13 at 16:22

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