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Find the real values of the parameter ‘a’ for which atleast one complex number z =x+iy satisfies both the equality | z- ai | = a+4 and the inequality | z-2 |

Find the real values of the parameter ‘a’ for which atleast one complex number z =x+iy satisfies both the equality
| z- ai | = a+4 and the inequality | z-2 |

Grade:12th pass

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago

Hello Student,
Please find the answer to your question below
[|z-ai| = a + 4]
[|z-2| < 1]
For these two to have atleast one solution, then both circles must intersect
[|x+iy-ai| = a+4]
[|x+i(y-a)| = a + 4]
[|x+i(y-a)|^{2} = (a + 4)^{2}]
[x^{2}+(y-a)^{2} = (a + 4)^{2}]
1st circle intersect the x – axis at 1 & 3.
2nd circle intersect x – axis
[x^{2}+(0-a)^{2} = (a + 4)^{2}]
[x^{2} + a^{2} = a^{2} + 16 + 8a]
[x^{2} = 16 + 8a]
16 + 8a = 1
so a=-15/8 and
if 16+8a=9
a=-7/8
hence -15/8<a<-7/8

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