If in a $\Delta A B C, \mathbf{O}$ and $\mathbf{O}^{\prime}$ are the incentre and orthocentre respectively, then $\left(\boldsymbol{O}^{\prime} \mathbf{A}+\boldsymbol{O}^{\prime} \mathbf{B}\right.$ $\left.+\mathbf{O}^{\prime} \mathbf{C}\right)$ is equal to

• Correct answer
  A

  $2 \mathbf{O}^{\prime} \mathbf{O}$
• B

  $0^{\prime} 0$
• C

  $00^{\prime}$
• D

  2 OO

Solution

O' $\mathbf{A}=\mathbf{O}^{\prime} \mathbf{O}+\mathbf{O A}$ $\mathbf{O}^{\prime} \mathbf{B}=\mathbf{O}^{\prime} \mathbf{O}+\mathbf{O B}$ $\mathbf{O}^{\prime} \mathbf{C}=\mathbf{O}^{\prime} \mathbf{O}+\mathbf{O C}$
$\Rightarrow \mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}$ $=3 \mathbf{0} \mathbf{0}+(\mathbf{O} \mathbf{A}+\mathbf{O B}+\mathbf{O C}) \ldots(\mathrm{i})$ $\because \quad \mathbf{O A}+\mathbf{O B}+\mathbf{O C}=\mathbf{O O}^{\prime}=-\mathbf{O}^{\prime} \mathbf{O}$ $\therefore \quad \mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}=3 \mathbf{O}^{\prime} \mathbf{O}-\mathbf{O}^{\prime} \mathbf{O}$ [from Eq. (i)] $\mathbf{O}^{\prime} \mathbf{A}+\mathbf{O}^{\prime} \mathbf{B}+\mathbf{O}^{\prime} \mathbf{C}=2 \mathbf{O}^{\prime} \mathbf{O}$

Hence, the correct answer is option: 'A'.