The equation of plane containing line x - y = 1, z = 1 and parallel...

Equation of plane containing line $x - y - 1 = 0$ and $z - 1 = 0$ is $( x - y - 1 ) + \lambda ( z - 1 ) = 0$

$⇒ x - y + \lambdaz - 1 - \lambda = 0... ( i )$

Since, this plane is parallel to line $\dfrac{x}{2} - \dfrac{z}{3} = 1$ and $y = 3$

$\dfrac{x - 2}{2} = \dfrac{y - 3}{0} = \dfrac{z}{3}... ( ii )$

$∴ 2 ( 1 ) + ( 0 ) ( - 1 ) + 3 ( \lambda ) = 0$

$⇒ \lambda = - 2 / 3$

Putting the value of $\lambda$ in Eq. $( i )$, we get

$x - y - \dfrac{2}{3}z - 1 + \dfrac{2}{3} = 0 ⇒ 3x - 3y - 2z = 1$