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Question

The equation |z+i||zi|=k represents a hyperbola, if

A
k(2,2)
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B
k(2,0)
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C
k{0,2}
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D
k{2,2}
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Solution

The correct option is A k(2,2)
Let z=x+iy
Therefore
|zi|+|z+i|=k

x2+(y1)2+x2+(y+1)2=k
x2+(y1)2=kx2+(y+1)2
x2+(y1)2=k2+x2+(y+1)22kx2+(y+1)2
2kx2+(y+1)2=k2+(y+1)2(y1)2
2kx2+(y+1)2=k2+4y
4k2(x2+(y+1)2)=k4+16y2+8k2y
4k2x2+4k2y2+4k2y+4k2=k4+16y2+8k2y
4k2x2+y2(4k216)4k2y+4k2k4=0
Therefore it represents a hyperbola if
4k216<0
Or
4k2<16
k2<4
|k|<2
2<k<2

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