The locus of the centre of a circle which touches the circle \[\left| {z - {z_1}} \right| = a\] and\[\left| {z - {z_2}} \right| = b\,\] externally
(\[z,\,{z_1}\,\& \,{z_2}\,\] are complex numbers) will be :
Solutions · 1
Official solution
Let the circle be \[\left| {z - {z_3}} \right| = r.\]
Then according to given conditions
\[\left| {{z_3} - {z_1}} \right| = r + a\] (Shown in the image)
and \[\left| {{z_3} - {z_2}} \right| = r + b.\] (Shown in the image)
Eliminating r, we get
\[\left| {{z_3} - {z_1}} \right| - \left| {{z_3} - {z_2}} \right| = a - b.\]
\[\therefore\] Locus of center z3 is
\[\left| {z - {z_1}} \right| - \left| {z - {z_2}} \right| = a - b\] = constant.
Definition of hyperbola says, when difference of distance between two points is constant from a particular point then that particular point will lie on a hyperbola.
Here distance of z1 from z3 is = r+a and distance of z2 from z3 is = r+b
Now their difference = (r+a) - (r+b) = a−b = a constant
\[\therefore\] Locus of z3 is a hyperbola.