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The locus of the centre of the circle which touch the circle |zz1|=a and |zz2|=b externally, |ab|<|z1z2| when z, z1, z2 are complex numbers will be

  1. an ellipse
  2. a circle
  3. hyperbola
  4. parabola

A
parabola
B
a circle
C
an ellipse
D
hyperbola
Solution
Verified by Toppr

Let variable circle be |zz0|=γ
Then, touches externally
|z0z1|=a+γ and |z0z2|=b+γ
eliminating r,
|z0z1||z0z1|=ab(1)
also given,
|z1z2|>|ab|(2) [if |z1z2|=ab locus is a line with z lying on line segment joining z1 & z2 for |z1z2|<|ab| no locus substituting |ab|=|z0z1||z0z1|
|z1z2|+|z0z1|<|z0z1|
conradicts property of trangle that sum of two sides > third side]
Thus (1)and (2) ensure locus is hyperbola

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