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Question

If the triangle formed by the complex coordinates A(z),B(2+3i),C(4+5i) which satisfy the relations |z(2+3i)|=|z(4+5i)| and |z(3+4i)|4, then


Correct Answer
A
ar. (ABC)max=42 sq. unit
Your Answer
B
ar. (ABC)max=4 sq. unit
Correct Answer
C
if area of ABC is maximum, then unequal angle is <π4
Your Answer
D
if area of ABC is maximum, then unequal angle is π4

Solution

The correct options are
A ar. (ABC)max=42 sq. unit
C if area of ABC is maximum, then unequal angle is <π4

As given |z(2+3i)|=|z(4+5i)|
AB=AC
Hence ABC will be isosceles triangle
and lenght of perpendicular from A on BC will be meet at mid point of BC
D=(z1+z2)/2=3+4i
Hence length of perpendicular
=|z(3+4i)|=4 (for maximum value)
So required maximum area 
=12×4×|z1z2|=42 sq. unit
Now let
2+3iz4+5iz=eiθtanθ/2=DCAD=122tanθ=427arg(2+3iz4+5iz)=θ<π4

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