If the roots of the equation $Z^3+i Z^2+2 i=0$ are the vertices of a triangle $A B C$. then that triangle $A…

If the roots of the equation $Z^3+i Z^2+2 i=0$ are the vertices of a triangle $A B C$. then that triangle $A B C$ is
  1. a right angled triangle
  2. an equilateral triangle
  3. an isosceles triangle
  4. a right angled isosceles triangle

Solution

Given, $Z^3+i Z^2+2 i=0$ $\begin{aligned} & \Rightarrow(Z-i)\left(Z^2+2 i Z-2\right)=0 \\ & Z=i \text { or } Z^2+2 i Z-2=0 \\ & \Rightarrow Z=\frac{-2 i \pm 2}{2} \Rightarrow z=-i+1,-i-1 \end{aligned}$
Let $A(0,1), B(1,-1), C(-1,-1)$ $A B=\sqrt{5} ; B C=2 ; A C=\sqrt{5}$
Since, $A B^2+A C^2 \neq B C^2$ and $A B=B C \neq B C$. So, $\triangle A B C$ is an isosceles triangle.

Asked in: AP EAMCET 2024 (23 May Shift 1)

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