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If the circle C1: x^2 + y^2 = 16 interse...

If the circle `C_1: x^2 + y^2 = 16` intersects another circle `C_2` of radius `5` in such a manner that,the common chord is of maximum length and has a slope equal to `3/4`, then the co-ordinates of the centre of `C_2` are:

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The correct Answer is:
`(-(9)/(5),(12)/(5))and((9)/(5),-(12)/(5))`
Given,
and let `C_(2):(x-h)^(2)+(y-k)^(2)=25`
`therefore` Equation of common chords is `S_(1)-S_(2)=0`
`therefore 2hx+2ky=(h^(2)+k^(2)-9)`
If p be the length of perpendicular on it from the centre
(0, 0) of `C_(1)` of radius 4, then `p=(h^(2)+k^(2)-9)/(sqrt(4h^(2)+4k^(2)))`.
Also, the length of the chord is
`2sqrt(r^(2)-p^(2))=2sqrt(4^(2)-p^(2))`
The chord will be of maximum length, if ` phi = 0` or
`h^(2)+k^(2)-9=0rArr+(16)/(9)h^(2)=9`
`rArr h=pm(9)/(5)`
`therefore k=pm(12)/(5)` Hence,centres ar `((9)/(5),(12)/(5)) and (-(9)/(5), (12)/(5))`
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