Question

The plane $XOZ$divides the join of $(1,-1,5)$ and $(2,3,4)$ in the ratio $\lambda :1$, then$\lambda$ is

• A. $-3$
• B. $3$
• C. $\frac{-1}{3}$
• D. $\frac{1}{3}$

Answer 1

The correct option is D

$\frac{1}{3}$

Solve for $\lambda$

Given, Points are $(1,-1,5)$ and $(2,3,4)$

We know that if the point$(x,y,z)$divides the line segment with points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ in the$m:n$ internally then,

$\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}\mathbf{)}\mathbf{=}\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n}\right)$

Using above formula the desired point is $\left(\frac{2\lambda +1}{\lambda +1},\frac{3\lambda -1}{\lambda +1},\frac{4\lambda +5}{\lambda +1}\right)$
Since, this point lies in $XOZ$ plane then its$y$-co-ordinate should be zero

$$\begin{gathered} \frac{3\lambda -1}{\lambda +1}=0 \\ \Rightarrow \lambda =\frac{1}{3} \end{gathered}$$

Hence, the correct option is (D).