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the area of triangle whose vertices are the roots z 3 +iz 2 +2i

the area of triangle whose vertices are the roots z3+iz2+2i

Question Image
Grade:12th pass

1 Answers

Arun
25763 Points
3 years ago
Dear Ashish
 

iven 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

 

Regards

Arun (askIITians forum expert)

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