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Algebra> Find the real values of the parameter ‘a’...

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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|

saurabh , 10 Years ago

Grade 12th pass

1 Answers

Jitender Singh

Last Activity: 10 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
$x^{2}+(0-a)^{2} = (a + 4)^{2}$
$x^{2} + a^{2} = a^{2} + 16 + 8a$
$x^{2} = 16 + 8a$
$16 + 8a = 1$
$a = \frac{-15}{8}$
$16 + 8a = 9$
$a = \frac{-7}{8}$
$\Rightarrow \frac{-15}{8}< a< \frac{-7}{8}$

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