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# A(z1),B(z2),C(z3) are 3 complex numbers forming a triangle on the circle |z|=1.The altitude from z1 meets the circle at Z(e) and p is the image of Z(e) with respect to the line BC and d is the mid point of the line PZ(e) then the complex number associated with d is...................A. Z1+2Z2+3Z3     B.Z1+Z2+Z3   C.Z1-Z2-Z3   D.Z1+Z2-Z3 11 years ago

Dear Naidu,

The point D is nothing but the point of intersection of AZ and BC.

Please check the options/question, the coordinates of P will be z1+z2+z3, coordinates of Z(e) will be (-z2z3/z1) and that of D will be 1/2(z1+z2+z3-z2z3/z1).

For Z:

Eqn of AZ:  (z-z1)/(z'-z1') + (z2-z3)/(z2'-z3')=0

So, (e-z1)/(e'-z1') + (z2-z3)/(z2'-z3')=0

|z1|=|z2|=|z3|=|e|=1 (they lie on circle),

so on simplifying using z1'=1/z1 etc => Z(e)=-z2z3/z1

For D:

BD is perpendicular to AD, so (z-z1)/(z'-z1') + (z-z2)/(z'-z2')=0.

Solve with  (z-z1)/(z'-z1') + (z2-z3)/(z2'-z3')=0 to get D as (1/2(z1+z2+z3-z2z3/z1)

For P:

use D is mid-pt of PZ(e) => P is z1+z2+z3

Hope this helps.

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