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Q.
If the lines $\frac{x - 1}{2}=\frac{y}{- 1}=\frac{z}{2}$ and $x-y+z-2=0=\lambda x+3z+5$ are coplanar, then the value of $7\lambda $ is equal to
NTA AbhyasNTA Abhyas 2020
Solution:
A plane passing through the line of intersection of the given $2$ planes is $\left(x - y + z - 2\right)+\mu \left(\lambda x + 3 z + 5\right)=0$
$\Rightarrow x\left(1 + \lambda \mu \right)-y+z\left(1 + 3 \mu \right)+5\mu -2=0$
Since the given line $\frac{x - 1}{2}=\frac{y}{- 1}=\frac{z}{2}$ must lie in this plane
$\Rightarrow 2\left(1 + \lambda \mu \right)+\left(- 1\right)\left(- 1\right)+\left(1 + 3 \mu \right)2=0$
$\Rightarrow 2+2\lambda \mu +1+2+6\mu =0$
$\Rightarrow 2\lambda \mu +6\mu +5=0\ldots ..\left(i\right)$
The point $\left(1 , 0 , 0\right)$ must satisfy the equation of the plane
$\Rightarrow 1+\lambda \mu -0+0+5\mu -2=0$
$\Rightarrow \lambda \mu +5\mu -1=0\ldots \ldots \left(i i\right)$
Solving $\left(i\right)$ and $\left(i i\right)$ we get,
$\mu =\frac{7}{4}, \, \lambda =\frac{- 31}{7}$
$\Rightarrow 7\lambda =-31$