If `||z-z_1|-||z-z_2||=k`, where `k<|z_1-z_2|` represents a hyperbola in argand plane then the equation of the asymptotes of the hyperbola are

Let z_(1) and z_(2) be two given complex numbers. The locus of z such that {:("Column -I", " Column -II"),( "(A) " |z-z_(1)|+|z-z_(2)| = " constant =k, where " k ne|z_(1)-z_(2)|, " (p) Circle with " z_(1) and z_(2) " as the vertices of diameter"),("(B)" |z-z_(1)|- |z-z_(2)|= " k where " k ne |z_(1)-z_(2)| ," (q) Circle "),("(C)"arg((z-z_(1))/(z-z_(2)))=+- pi/2 , " (r) Hyperbola "),("(D) If "omega" lies on " |omega| = 1 " then " , " (s) Ellipse"):}

Let z_(1) and z_(2) be two given complex numbers. The locus of z such that {:("Column -I", " Column -II"),( "(A) " |z-z_(1)|+|z-z_(2)| = " constant =k, where " k ne|z_(1)-z_(2)|, " (p) Circle with " z_(1) and z_(2) " as the vertices of diameter"),("(B)" |z-z_(1)|- |z-z_(2)|= " k where " k ne |z_(1)-z_(2)| ," (q) Circle "),("(C)"arg((z-z_(1))/(z-z_(2)))=+- pi/2 , " (r) Hyperbola "),("(D) If "omega" lies on " |omega| = 1 " then " 2007/omega " lies on " , " (s) Ellipse"):}

A(z_(1)) and B(z_(2)) are two fixed points in the Argand plane and P(z) is variable point satisfying |z-z_(1)|=k|z-z_(2)| , where k gt 0 and k ne 1 . The locus of is

If the equation |z-z_1|^2 + |z-z_2|^2 =k , represents the equation of a circle, where z_1 = 3i, z_2 = 4+3i are the extremities of a diameter, then the value of k is

If the equation |z-z_1|^2 + |z-z_2|^2 =k , represents the equation of a circle, where z_1 = 3i, z_2 = 4+3i are the extremities of a diameter, then the value of k is

Prove that |z-z_1|^2+|z-z_2|^2 = k will represent a circle if |z_1-z_2|^2 le 2k.

Prove that |z-z_1|^2+|z-z_2|^2=k will represent a real circle with center ((z_1+z_2)/2) on the Argand plane if 2kgeq|z_1-z_2|^2

How does |z-z1|-|z-z2||=K represents hyperbola

If the equation |z-z_(1)|^(2)+|z-z_(2)|^(2)=k represents the equation of a circle,where z_(1)=2+3 iota,z_(2)=4+3t are the extremities of a diameter,then the value of k is

If k gt 0, k ne 1, and z_(1), z_(2) in C , then |(z-z_(1))/(z-z_(2))| = k represents