Question:
if the roots of z^3+iz^2+2i=0 represents the vertices of any triangle in the argand plane then the area of triangle is ?
Answer:
Given cubic equation is
z3 + iz2 + 2i = 0
Now put z = i in LHS
i3 + i*i2 + 2i
= i*i2 + i*i2 + 2i
= -i -i +2i (since i2 = -1)
= -2i + 2i
= 0
= RHS
So z =i is a factor of given eqaution
Now when we divide z3 + iz2 + 2i by z-i, we get other two factors 1-i and -1-i.
So three roots are i , 1-i, -1-i.
Now vertices of triangle are (0, 1), (1, -1), (-1, -1).
Now are of triangle is defined as
|0 1 1 |
(1/2)* |1 -1 1|
|-1 -1 1|
= (1/2)({0 -1(1+1) + 1(-1-1)}
= (1/2)*(-2-2)
= -4/2
= -2
By taking positive sign
Area of the traingle is 2 unit2
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