Question
Question: If \(z_{1},z_{2},z_{3}\) are the vertices of an equilateral triangle with \(z_{0}\) as its circumcen...
If z1,z2,z3 are the vertices of an equilateral triangle with z0 as its circumcentre then changing origin to z0, then (where z1,z2,z3 are new complex numbers of the vertices)
z12+z22+z32=0
z1z2+z2z3+z3z1=0
Both (1) and (2)
None of these
z12+z22+z32=0
Solution
Sol. In an equilateral triangle the circumcentre and the centroid are the same point. So,
z0=3z1+z2+z3⇒ z1+z2+z3=3z0 ..... (i)
To shift the origin at z0, we have to replace z1,z2,z3 and z0 by z1+z0,z2+z0,z3+z0and 0+z0 then equation (i) becomes (z1+z0)+(z2+z0)+(z3+z0)=3(0+z0) ⇒ z1+z2+z3=0
On squaring z12+z22+z32+2(z1z2+z2z3+z3z1)=0
But triangle with vertices z1,z2 and z3 is equilateral, then z12+z22+z32=z1z2+z2z3+z3z1 .....(iii)
From (ii) and (iii) we get, 3(Z12+Z22+Z32)=0. Therefore, Z12+Z22+Z32=0.