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Question

If the circle x2+y26x10y+c=0 does not touch or intersect the coordinate axes and (1,4) lies inside the circle, then the number of integral values of c is

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Solution

x2+y26x10y+c=0
Centre is (3,5) and radius =32+52c=34c
34c>0c<34 (1)

Since, P(1,4) lies inside the circle,
CP<r
(31)2+(54)2<34c
c<29 (2)
Also, since the circle does not touch or intersect the coordinate axes, we have
r<5 and r<3
r<3
34c<3
c>25 (3)

From (1),(2) and (3),
25<c<29
Possible integral value of c is 26,27,28

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