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 \[\left| {{z}_{1}}-{{z}_{2}} \right|>\left| a-b \right|\] | |||
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