1
Question
The locus of the centre of the circle which touch the circle |z−z1|=a and |z−z2|=b externally, |a−b|<|z1−z2| when z, z1, z2 are complex numbers will be
The locus of the centre of the circle which touch the circle |z−z1|=a and |z−z2|=b externally, |a−b|<|z1−z2| when z, z1, z2 are complex numbers will be
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Solution
The correct option is C hyperbola
Let variable circle be |z−z0|=γ
Then, touches externally
|z0−z1|=a+γ and |z0−z2|=b+γ
eliminating r,
|z0−z1|−|z0−z1|=a−b−−(1)
also given,
|z1−z2|>|a−b|−−(2) [if |z1−z2|=a−b locus is a line with z lying on line segment joining z1 & z2 for |z1−z2|<|a−b| no locus substituting |a−b|=|z0−z1|−|z0−z1|
⇒|z1−z2|+|z0−z1|<|z0−z1|
conradicts property of trangle that sum of two sides > third side]
Thus (1)and (2) ensure locus is hyperbola
Let variable circle be |z−z0|=γ
Then, touches externally
|z0−z1|=a+γ and |z0−z2|=b+γ
eliminating r,
|z0−z1|−|z0−z1|=a−b−−(1)
also given,
|z1−z2|>|a−b|−−(2) [if |z1−z2|=a−b locus is a line with z lying on line segment joining z1 & z2 for |z1−z2|<|a−b| no locus substituting |a−b|=|z0−z1|−|z0−z1|
⇒|z1−z2|+|z0−z1|<|z0−z1|
conradicts property of trangle that sum of two sides > third side]
Thus (1)and (2) ensure locus is hyperbola
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