Question

# If the roots of $$\displaystyle\ z^{3}+iz^{2}+2i=0$$ represents the vertices of a $$\displaystyle\ \Delta ABC$$ in the Argand plane then the area of the triangle is

A
372
B
374
C
2
D
none of these

Solution

## The correct option is D $$\displaystyle\ 2$$One obvious root is $$i$$Then $$(z-i)(z^{2}+2iz-2)=0$$$$z=i$$And $$z^{2}+2iz-2=0$$$$(z+(i+1))(z+(i-1))=0$$$$z=-i-1$$ and $$z=-i+1$$Hence the triangle is formed by the vertices $$A(0,1),B(-1,-1),C(1,-1)$$Hence this is an isosceles triangle. with $$AB=AC$$Considering $$BC$$ as the base.The vertical height will be given by the distance between A(0,1) and the midpoint of BCHence $$H=2$$And Base$$=BC=2$$Hence area $$=\dfrac{B\times H}{2}$$$$=2$$.Maths

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