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Question: The plane \(XOZ\) divides the join of \(\left( 1,-1,5 \right)\) and \(\left( 2,3,4 \right)\) in the ...

The plane XOZXOZ divides the join of (1,1,5)\left( 1,-1,5 \right) and (2,3,4)\left( 2,3,4 \right) in the ratio λ:1\lambda :1, then λ\lambda is
A. -3
B. 14\dfrac{1}{4}
C. 3
D. 13\dfrac{1}{3}

Explanation

Solution

Hint: We know that in XOZXOZ plane, we have the y coordinates as equal to 0. We will find the value of λ\lambda using this concept as y=oy=o divides the join of (1,1,5)\left( 1,-1,5 \right) and (2,3,4)\left( 2,3,4 \right) in the ratio λ:1\lambda :1, so we get, 3λ1λ+1=0\dfrac{3\lambda -1}{\lambda +1}=0.

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


Now we will find the coordinates of XX and ZZ. We know that for any two points, (x1,x2)\left( {{x}_{1}},{{x}_{2}} \right) and (y1,y2)\left( {{y}_{1}},{{y}_{2}} \right), when they are divided by any point in the ratio of m:nm:n, then they can be written as, (m1n)=x2(m)+x1(n)m+n,y2(m)+y1(n)m+n\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 (X,Z)\left( X,Z \right) because the point, y=oy=o. So, we can write it as,
3(λ)+(1)(1)λ+1=0 3λ1λ+1=0 3λ1=0 3λ=1 λ=13 \begin{aligned} & \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{aligned}
Thus, the value of λ\lambda is 13\dfrac{1}{3} and the plane XOZXOZ divides the point in the ratio of 13:1\dfrac{1}{3}:1.
Therefore, the correct answer to the given question is option D, 13\dfrac{1}{3}.

Note: In this type of questions, the students usually find the line using the two given points, (x1,x2,x3)\left( {{x}_{1}},{{x}_{2}},{{x}_{3}} \right) and (y1,y2,y3)\left( {{y}_{1}},{{y}_{2}},{{y}_{3}} \right), then they use the ratio λ:1\lambda :1 to divide and find the value of λ\lambda , but that is quite a long method. On the other hand, if we apply only a single concept, that is, in a plane (X,Z)\left( X,Z \right), the coordinates of y=oy=o, we can directly find the value of λ\lambda as we have done in the question.