Question
If $$P,P'$$ represent the complex number $${z_1}$$ and its additive inverse respectively then the complex equation of the circle with $$PP'$$ as a diameter is
A.
$$\frac{z}{{{z_1}}} = \left( {\frac{{{{\overline z }_1}}}{z}} \right)$$
B.
$$z\overline z + {z_1}{\overline z _1} = 0$$
C.
$$z{\overline z _1} + \overline z {z_1} = 0$$
D.
None of these
Answer :
$$\frac{z}{{{z_1}}} = \left( {\frac{{{{\overline z }_1}}}{z}} \right)$$
Solution :
Clearly, $$\left| z \right| = \left| {{z_1}} \right|$$
$$\eqalign{ & \therefore \,\,z\overline z = {z_1}{\overline z _1} \cr & {\text{or, }}\frac{z}{{{z_1}}} = \frac{{{{\overline z }_1}}}{{\overline z }} = \overline {\left( {\frac{{{z_1}}}{z}} \right)} . \cr} $$
Clearly, $$\left| z \right| = \left| {{z_1}} \right|$$
$$\eqalign{ & \therefore \,\,z\overline z = {z_1}{\overline z _1} \cr & {\text{or, }}\frac{z}{{{z_1}}} = \frac{{{{\overline z }_1}}}{{\overline z }} = \overline {\left( {\frac{{{z_1}}}{z}} \right)} . \cr} $$