Question

The plane $XOZ$ divides the join of $\left( 1,-1,5 \right)$ and $\left( 2,3,4 \right)$ in the ratio $\lambda :1$, then $\lambda $ is
A. -3
B. $\dfrac{1}{4}$
C. 3
D. $\dfrac{1}{3}$

Answer 1

Hint: We know that in $XOZ$ plane, we have the y coordinates as equal to 0. We will find the value of $\lambda $ using this concept as $y=o$ divides the join of $\left( 1,-1,5 \right)$ and $\left( 2,3,4 \right)$ in the ratio $\lambda :1$, so we get, $\dfrac{3\lambda -1}{\lambda +1}=0$.

Complete step-by-step solution -
It has been given in the question that the plane $XOZ$ divides the join of $\left( 1,-1,5 \right)$ and $\left( 2,3,4 \right)$ in the ratio $\lambda :1$ and we have to find the value of $\lambda $. We know that in the $XZ$ plane, the coordinates of y axis is 0, or we get, $y=o$. We also know that $XOZ$ divides the join of $\left( 1,-1,5 \right)$ and $\left( 2,3,4 \right)$ in the ratio $\lambda :1$.

Now we will find the coordinates of $X$ and $Z$. We know that for any two points, $\left( {{x}_{1}},{{x}_{2}} \right)$ and $\left( {{y}_{1}},{{y}_{2}} \right)$, when they are divided by any point in the ratio of $m:n$, then they can be written as, $\left( {{m}_{1}}n \right)=\dfrac{{{x}_{2}}\left( m \right)+{{x}_{1}}\left( n \right)}{m+n},\dfrac{{{y}_{2}}\left( m \right)+{{y}_{1}}\left( n \right)}{m+n}$ .
But here we have to find $\left( X,Z \right)$ because the point, $y=o$. So, we can write it as,
$\begin{align}
  & \dfrac{3\left( \lambda \right)+\left( -1 \right)\left( 1 \right)}{\lambda +1}=0 \\
 & \Rightarrow \dfrac{3\lambda -1}{\lambda +1}=0 \\
 & \Rightarrow 3\lambda -1=0 \\
 & \Rightarrow 3\lambda =1 \\
 & \Rightarrow \lambda =\dfrac{1}{3} \\
\end{align}$
Thus, the value of $\lambda $ is $\dfrac{1}{3}$ and the plane $XOZ$ divides the point in the ratio of $\dfrac{1}{3}:1$.
Therefore, the correct answer to the given question is option D, $\dfrac{1}{3}$.

Note: In this type of questions, the students usually find the line using the two given points, $\left( {{x}_{1}},{{x}_{2}},{{x}_{3}} \right)$ and $\left( {{y}_{1}},{{y}_{2}},{{y}_{3}} \right)$, then they use the ratio $\lambda :1$ to divide and find the value of $\lambda $, but that is quite a long method. On the other hand, if we apply only a single concept, that is, in a plane $\left( X,Z \right)$, the coordinates of $y=o$, we can directly find the value of $\lambda $ as we have done in the question.