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Question

If the equation x2+y2+2λx+4=0 and x2+y2+4λy+8=0 represent real circles then the value of λ can be

A
5
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B
2
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C
3
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D
All of these
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Solution

The correct option is D All of these
Solution:- (D) all of these
C1:x2+y2+2λx+4=0
x2+2λx+λ2λ2+y2+4=0
(x+λ)2+y2λ2+4=0
(x+λ)2+y2=λ24
r12=λ240
(λ2)(λ+2)0
λ(,2][2,).....(1)
C2:x2+y2+4λy+8=0
x2+y2+4λy+4λ24λ2+8=0
x2+(y+2λ)24λ2+8=0
x2+(y+2λ)2=4λ28
r22=4λ280
λ220
λ(,2][2,).....(2)
From eqn(1)&(2), we get
λ(1)(2)
λ(,2][2,)
Since all of the given values lie within the range.
Hence the correct answer is (D) all of these.

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