Search any question & find its solution
Question:
Answered & Verified by Expert
The incentre of the triangle $\mathrm{ABC}$, whose vertices are $\mathrm{A}(0,2,1), \mathrm{B}(-2,0,0)$ and $\mathrm{C}(-2,0,2)$, is
Options:
Solution:
2759 Upvotes
Verified Answer
The correct answer is:
$\left(-\frac{3}{2}, \frac{1}{2}, 1\right)$
Let $\bar{a}=2 \hat{j}+\hat{k}, \hat{b}=-2 \hat{i}, \hat{c}=-2 \hat{i}+2 \hat{k}$
$\therefore \quad \overline{\mathrm{AB}}=-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$,
$\overline{\mathrm{BC}}=2 \hat{\mathrm{k}}$
$\begin{aligned} & \overline{\mathrm{AC}}=-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \\ & \Rightarrow|\overline{\mathrm{AB}}|=3,|\overline{\mathrm{BC}}|=2,|\overline{\mathrm{AC}}|=3\end{aligned}$
Incentre of $\triangle \mathrm{ABC}$ is given by
$\frac{|\overline{\mathrm{AB}}| \overline{\mathrm{c}}+|\overline{\mathrm{BC}}| \overline{\mathrm{a}}+|\overline{\mathrm{AC}}| \overline{\mathrm{b}}}{|\overline{\mathrm{AB}}|+|\overline{\mathrm{BC}}|+|\overline{\mathrm{AC}}|}$
$\begin{aligned} & =\frac{3(-2 \hat{i}+2 \hat{k})+2(2 \hat{j}+\hat{k})+3(-2 \hat{i})}{3+2+3} \\ & =\frac{-12 \hat{i}+4 \hat{j}+8 \hat{k}}{8} \\ & =-\frac{3}{2} \hat{i}+\frac{1}{2} \hat{j}+\hat{k}\end{aligned}$
$\therefore \quad \overline{\mathrm{AB}}=-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$,
$\overline{\mathrm{BC}}=2 \hat{\mathrm{k}}$
$\begin{aligned} & \overline{\mathrm{AC}}=-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \\ & \Rightarrow|\overline{\mathrm{AB}}|=3,|\overline{\mathrm{BC}}|=2,|\overline{\mathrm{AC}}|=3\end{aligned}$
Incentre of $\triangle \mathrm{ABC}$ is given by
$\frac{|\overline{\mathrm{AB}}| \overline{\mathrm{c}}+|\overline{\mathrm{BC}}| \overline{\mathrm{a}}+|\overline{\mathrm{AC}}| \overline{\mathrm{b}}}{|\overline{\mathrm{AB}}|+|\overline{\mathrm{BC}}|+|\overline{\mathrm{AC}}|}$
$\begin{aligned} & =\frac{3(-2 \hat{i}+2 \hat{k})+2(2 \hat{j}+\hat{k})+3(-2 \hat{i})}{3+2+3} \\ & =\frac{-12 \hat{i}+4 \hat{j}+8 \hat{k}}{8} \\ & =-\frac{3}{2} \hat{i}+\frac{1}{2} \hat{j}+\hat{k}\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.