The equation of plane equidistant from lines x−11=y−22=z−33 and 2x−y−4=0=3x−z−8 is/are

Since, line L1 II L2. Hence, there will be infinite planes cutting line L and parallel to given line.Equation of line L:x−121=y+12=z+523⇒ Plane P1:2x−y−2=0Plane P2:3x−z−4=0⇒ Required plane p:p1+λp2=0 (2x−y−2)+λ(3x−z−4)=0,λ∈RAnother is system of planes parallel to plane containing L1 and L2Normal to lines L1 and L2:x→=|iˆjˆkˆ1611123|=−4iˆ+8jˆ−4kˆ⇒ One of D.R's normal =(1,−2,1)⇒ Equation of plane containing line's L1 and L2x−2y+z=0⇒ Any plane parallel to the plane is x−2y+z=d,d∈R

The equation of plane equidistant from lines x−11=y−22=z−33 and 2x−y−4=0=3x−z−8 is/are

Since, line L1 II L2. Hence, there will be infinite planes cutting line L and parallel to given line.Equation of line L:x−121=y+12=z+523⇒ Plane P1:2x−y−2=0Plane P2:3x−z−4=0⇒ Required plane p:p1+λp2=0 (2x−y−2)+λ(3x−z−4)=0,λ∈RAnother is system of planes parallel to plane containing L1 and L2Normal to lines L1 and L2:x→=|iˆjˆkˆ1611123|=−4iˆ+8jˆ−4kˆ⇒ One of D.R's normal =(1,−2,1)⇒ Equation of plane containing line's L1 and L2x−2y+z=0⇒ Any plane parallel to the plane is x−2y+z=d,d∈R

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The equation of plane equidistant from lines $\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$ and $2 x - y - 4 = 0 = 3 x - z - 8$ is/are

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$x - 2 y + z = 5$

$x + y - z - 2 = 0$

$2 x - 4 y + 2 z = 3 p \forall p \in R$

$(2 + 3 λ) x - y - λ z - 2 (2 + λ) = 0 \forall λ \in R$

answer is A, B, D.

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Detailed Solution

Since, line $L_{1}$ II $L_{2}$ . Hence, there will be infinite planes cutting line L and parallel to given line.
Equation of line $L : \frac{x - \frac{1}{2}}{1} = \frac{y + 1}{2} = \frac{z + \frac{5}{2}}{3}$
$\Rightarrow$ Plane $P_{1} : 2 x - y - 2 = 0$
Plane $P_{2} : 3 x - z - 4 = 0$
$\Rightarrow$ Required plane $p : p_{1} + λ p_{2} = 0$
$(2 x - y - 2) + λ (3 x - z - 4) = 0, λ \in R$

Another is system of planes parallel to plane containing $L_{1}$ and $L_{2}$
Normal to lines $L_{1}$ and $L_{2} : \vec{x} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 6 & 11 \\ 1 & 2 & 3 \end{matrix} | = - 4 \hat{i} + 8 \hat{j} - 4 \hat{k}$
⇒ One of D.R's normal $= (1, - 2, 1)$
⇒ Equation of plane containing line's $L_{1}$ and $L_{2} x - 2 y + z = 0$
⇒ Any plane parallel to the plane is $x - 2 y + z = d, d \in R$