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Question: The plane \[XOZ\] divides the joint of \[\left( {1, - 1,5} \right)\] and \[\left( {2,3,4} \right)\] ...

The plane XOZXOZ divides the joint 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) −3-3
B) 14\dfrac{1}{4} ​
C) 3
D) 13\dfrac{1}{3}

Explanation

Solution

Here we have to find the value of the variable. For that, we will use the section formula to find the coordinates of a point at which the given plane divide the line joining the points (1,−1,5)\left( {1, - 1,5} \right) and (1,−1,5)\left( {1, - 1,5} \right). The obtained point will lie in the x-z plane and thus, the y-coordinate of the point will be zero. We will equate the y-coordinate of the obtained point with zero. From there, we will get the value of the required variable.

Complete step by step solution:
We will use the section formula to find the coordinates of a point at which the given plane divides the line joining the points (1,−1,5)\left( {1, - 1,5} \right) and (2,3,4)\left( {2,3,4} \right).
Therefore, the desired point is (2λ+1λ+1+3λ−1λ+1+4λ+5λ+1)\left( {\dfrac{{2\lambda + 1}}{{\lambda + 1}} + \dfrac{{3\lambda - 1}}{{\lambda + 1}} + \dfrac{{4\lambda + 5}}{{\lambda + 1}}} \right) .
Since, the point line in the plane XOZXOZ , thus, the y-coordinate of the point is zero.
⇒3λ−1λ+1=0\Rightarrow \dfrac{{3\lambda - 1}}{{\lambda + 1}} = 0
On cross multiplying the terms, we get
⇒3λ−1=0\Rightarrow 3\lambda - 1 = 0
Adding 1 to both sides, we get
⇒3λ−1+1=0+1 ⇒3λ=1\begin{array}{l} \Rightarrow 3\lambda - 1 + 1 = 0 + 1\\\ \Rightarrow 3\lambda = 1\end{array}
Dividing both sides by 3, we get
⇒λ=13\Rightarrow \lambda = \dfrac{1}{3}

Thus, the correct option is option D.

Note:
Here we have obtained the point using the section formula. Section formula is a formula used to find the coordinates of a point which divides the line joining two points in a ratio, either internally or externally. As the given plane divides the line joining the two given points here, the plane is defined as a flat and two dimensional infinite surface.
Some important properties of plane are as follows:-
If two distinct lines are perpendicular to the same plane then these two lines will be parallel to each other.
If two distinct planes are perpendicular to the same line then these two planes will be parallel to each other.