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Question

If the circle x2+y2āˆ’6xāˆ’10y+k=0 does not touch or intersect the coordinate axes, and the point (1,4) is inside the circle, then the range of k is

A
25<k<29
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B
9<k<29
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C
9<k<25
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D
5<k<25
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Solution

The correct option is A 25<k<29

The equation of the circle is x2+y26x10y+k=0
Centre, C(3,5)
Radius, r=32+52k=34k
Since the circle does not touch or intersect the xaxis,
r<ycoordinate of centre C
or, 34k<5
34k<25
k>9 (1)

Also, given circle does not touch or intersect the yaxis.
r<xcoordinate of centre C
or, 34k<3
34k<9
k>25 (2)

Since point (1,4) is inside the circle, then its distance from centre C<r,
or, (31)2+(54)2<34k
5<34k
k<29 (3)
Taking intersection of (1),(2),(3),
Range of k is 25<k<29.

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