Reflection of the line (x - 1)/(- 1) = (y - 2)/(3) = (z - 4)/(1) in...

Given line $\dfrac{x - 1}{- 1} = \dfrac{y - 2}{3} = \dfrac{z - 4}{1}$ is passes through $P ( 1, 2, 4 )$.

To find the reflection of line we need one more point on the line clearly $M ( 0, 5, 5 )$ also lie on line.

Let $N ( \alpha, \beta, \gamma )$ be the reflection $M$ in the $x + y + z = 7$

$∴ \dfrac{\alpha}{2} + \dfrac{\beta + 5}{2} + \dfrac{\gamma + 5}{2} = 7$

$⇒ \alpha + \beta + \gamma = 4$

Also, $MN$ is perpendicular, i.e. parallel to the normal of the plane

$∴ \dfrac{\alpha}{1} = \dfrac{\beta - 5}{1} = \dfrac{\gamma - 5}{1} = \lambda$

$⇒ \alpha = \lambda, \beta = \lambda + 5, \gamma = \lambda + 5$

$∴ \lambda + \lambda + 5 + \lambda + 5 = 4$

$⇒ \lambda = - 2$

$∴ N = ( - 2, 3, 5 )$

Equation $PN$ is $\dfrac{x - 1}{- 3} = \dfrac{y - 2}{1} = \dfrac{z - 4}{- 1}$