# If L1 is the line of intersection of the planes 2x−2y+3z−2=0, x−y+z+1=0 and L2 is the line of intersection of the planes x+2y−z−3=0, 3x−y+2z−1=0, then the distance of the origin from the plane, containing the lines L1 and L2, is : Option 1) $\frac{1}{\sqrt{2}}$ Option 2) $\frac{1}{4\sqrt{2}}$ Option 3) $\frac{1}{3\sqrt{2}}$ Option 4) $\frac{1}{2\sqrt{2}}$

H Himanshu

$p_{1}+\lambda p_{2}=0$

$(2x-2y+3z-2)+\lambda (x-y+z+1)=0$

$(2+\lambda )x+(-2-\lambda )y+ (3+\lambda )z- (2-\lambda )=0$

also $L_{2}$  has two planes

$\begin{vmatrix} (2+\lambda ) & -2-\lambda & 3+\lambda \\ 1&2 &-1 \\ 3 &-1 &2 \end{vmatrix}=0$

$\lambda =5$

Equation of any plane passing through the line of intersection of two planes (Cartesian form ) -

The equation of any plane passing through the line of intersection of two planes

$ax+by+cz+d= 0$ and

$a_{1}x+b_{1}y+c_{1}z+d_{1}= 0$ is given by

$\left ( ax+by+cz+d \right )+\lambda \left ( a_{1}x+b_{1}y+c_{1}z+d _{1}\right )= 0$

-

Option 1)

$\frac{1}{\sqrt{2}}$

This is incorrect

Option 2)

$\frac{1}{4\sqrt{2}}$

This is incorrect

Option 3)

$\frac{1}{3\sqrt{2}}$

This is correct

Option 4)

$\frac{1}{2\sqrt{2}}$

This is incorrect

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