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Question
The locus of the centre of a circle which touches the circles |z−z1|=a and |z−z2|=b externally where z,z1, and z2 are complex numbers is
The locus of the centre of a circle which touches the circles |z−z1|=a and |z−z2|=b externally where z,z1, and z2 are complex numbers is
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Solution
The correct option is B a hyperbola
Let the circle |z-c|=d touch both the circles. We have to find the locus of the complex number c.
We have
|c−z1|=a+dand⇒|c−z2|=b+d⇒|c−z1|−|c−z2|=a−b⇒|z−z1|−|z−z2|=a−b
Thus difference between the distance of c from z1 and z2 is constant.
Thus the locus of c is a hyperbola with foci z1 and z2.
Let the circle |z-c|=d touch both the circles. We have to find the locus of the complex number c.
We have
|c−z1|=a+dand⇒|c−z2|=b+d⇒|c−z1|−|c−z2|=a−b⇒|z−z1|−|z−z2|=a−b
Thus difference between the distance of c from z1 and z2 is constant.
Thus the locus of c is a hyperbola with foci z1 and z2.
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