Question

Let points $A(3,4,-5),B(1,-2,3)$ and a point $C(x,y,z)$ lies on $\overrightarrow{AB}.$ If $x=y$ then $\overrightarrow{OC}$ divides $\overrightarrow{AB}$ in ratio :

• A. internally in $1:2$
• B. internally in $2:1$
• C. internally in $1:3$
• D. internally in $3:1$

Answer 1

The correct option is C internally in $1:3$

Given points : points $A(3,4,-5),B(1,-2,3)$
let $\overrightarrow{OC}$ divides $\overrightarrow{AB}$ in $\lambda :1,$
$\Rightarrow \overrightarrow{OC}=\frac{\overrightarrow{OA}+\lambda \overrightarrow{OB}}{\lambda +1}\Rightarrow x\hat{i}+y\hat{j}+z\hat{k}=\frac{3\hat{i}+4\hat{j}-5\hat{k}+\lambda (\hat{i}-2\hat{j}+3\hat{k})}{\lambda +1}=\frac{(3+\lambda )\hat{i}+(4-2\lambda )\hat{j}+(3\lambda -5)\hat{k}}{\lambda +1}$
As, $x=y$
$\Rightarrow 3+\lambda =4-2\lambda \Rightarrow 3\lambda =1$