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Question

If the circles x2+y216x20y+164=r2 and (x4)2+(y7)2=36 intersect at two distinct points, then :

A
1<r<11
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B
0<r<1
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C
r=11
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D
r>11
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Solution

The correct option is A 1<r<11
x2+y216x20y+164=r2
(x8)2+(y10)2=r2
(x4)2+(y7)2=36=62

The distance between their centres is,
C1C2=(84)2+(107)2
C1C2=5
As we know,
|r6|<C1C2<r+6
|r6|<5<|r+6|

If |r6|<5
r6<5 and r6>5
r<11 and r>1
r(1,11) ...(1)

and |r+6|>5
r+6>5 and r+6<5
r (,11)(1,) ...(2)
From (1) and (2),
r (1,11)

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