Circles of radius 5 units intersects the circle `(x-1)^(2)+(x-2)^(2)=9` in a such a way that the length of the common chord is of maximum length. If the slope of common chord is `(3)/(4)`, then find the centre of the circle.

Answer

`((17)/(5),(-6)/(5))` and `((-7)/(5),(26)/(5))`

Explanation

Clearly, common chord AB will be along the diameter of given circle.
In triangle `AC_(2)C_(2),AC_(1)=3,AC_(2)=5`.
`:. C_(1)C_(2)=C_(1)C_(3)=4`.
Given that slope of AB is `(3)/(4)`. `:. `Slope of `C_(2)C_(3), -(4)/(3)= tan theta`
Using parametric form of straight line, coordinates of `C_(2)` and `C_(3)` are given by
`(1+-4 cos theta, 2overset(-)(+)4 sin theta)-= (1+- 4 (3)/(5), 2 overset(-)(+)4(4)/(5))`
`-= ((17)/(5),(6)/(5))` and `((-7)/(5),(26)/(5))`