question_answer

The locus of the centre of a circle, which touches the circles \[\left| z-{{z}_{1}} \right|=a\] and \[\left| z-{{z}_{2}} \right|=b\] externally can be

A) an ellipse

B) a hyperbola

C) a circle

D) parabola

Correct Answer: B

Solution :

Let the Centre of the circle be \[{{z}_{0}}\]and radius \[r.\]then its Equation is \[\left| z-{{z}_{0}} \right|=r\]			?..(1)
Then circle (1) touches the circle \[\left| z-{{z}_{1}} \right|=a\]externally
\[\therefore \] Distance between Centre?s=sum of radii\[\Rightarrow \left| {{z}_{0}}-{{z}_{1}} \right|=a+r\]		? (2)
Similarly \[\Rightarrow \left| {{z}_{0}}-{{z}_{2}} \right|=b+r\]	? (3)
Subtract (2) and (3),
\[\left| {{z}_{0}}-{{z}_{1}} \right|-\left| {{z}_{0}}-{{z}_{2}} \right|=a-b\]
\[\therefore \]\[{{z}_{0}}\]Lies on the curve
\[\left| z-{{z}_{1}} \right|-\left| z-{{z}_{2}} \right|=a-b,\] Which is equation of hyperbola, provided \[\left| {{z}_{1}}-{{z}_{2}} \right|>\left| a-b \right|\]