Search any question & find its solution
Question:
Answered & Verified by Expert
If $A(0,4,0), \quad B(0,0,3)$ and $C(0,4,3)$ are the vertices of $\Delta A B C$, then its
incentre is
Options:
incentre is
Solution:
2084 Upvotes
Verified Answer
The correct answer is:
(0,3,2)
Let
$$
\begin{array}{ll}
\overline{\mathrm{a}}=4 \hat{\mathrm{j}} & \therefore|\overline{\mathrm{a}}|=\mathrm{a}=4 \\
\overline{\mathrm{b}}=3 \hat{\mathrm{k}} & \therefore|\overline{\mathrm{b}}|=\mathrm{b}=3 \\
\overline{\mathrm{c}}=4 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} & \therefore|\overline{\mathrm{c}}|=\mathrm{c}=\sqrt{16+9}=5
\end{array}
$$
Let $H(\bar{h})$ be the incentre
$$
\begin{array}{l}
\text { Incentre is given by } \frac{a \bar{a}+b \bar{b}+c \bar{c}}{a+b+c}=\bar{h} \\
\Rightarrow \bar{h}=\frac{4(4 \hat{j})+3(3 \hat{k})+5(4 \hat{j}+3 \hat{k})}{4+3+5} \\
\quad=\frac{16 \hat{j}+9 \hat{k}+20 \hat{j}+15 \hat{k}}{12}=\frac{36 \hat{j}+24 \hat{k}}{12}=3 \hat{j}+2 \hat{k}
\end{array}
$$
Thus co-ordinate of incentre is $(0,3,2)$
$$
\begin{array}{ll}
\overline{\mathrm{a}}=4 \hat{\mathrm{j}} & \therefore|\overline{\mathrm{a}}|=\mathrm{a}=4 \\
\overline{\mathrm{b}}=3 \hat{\mathrm{k}} & \therefore|\overline{\mathrm{b}}|=\mathrm{b}=3 \\
\overline{\mathrm{c}}=4 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} & \therefore|\overline{\mathrm{c}}|=\mathrm{c}=\sqrt{16+9}=5
\end{array}
$$
Let $H(\bar{h})$ be the incentre
$$
\begin{array}{l}
\text { Incentre is given by } \frac{a \bar{a}+b \bar{b}+c \bar{c}}{a+b+c}=\bar{h} \\
\Rightarrow \bar{h}=\frac{4(4 \hat{j})+3(3 \hat{k})+5(4 \hat{j}+3 \hat{k})}{4+3+5} \\
\quad=\frac{16 \hat{j}+9 \hat{k}+20 \hat{j}+15 \hat{k}}{12}=\frac{36 \hat{j}+24 \hat{k}}{12}=3 \hat{j}+2 \hat{k}
\end{array}
$$
Thus co-ordinate of incentre is $(0,3,2)$
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.