Reflection of the line $\frac{x-1}{-1}=\frac{y-2}{3}=\frac{z-4}{1}$ in the plane $x+y+z=7$ is

Note that A (1, 2, 4) also lies on the plane. P (0, 5, 5) is a point on the line. Find its reflection $Q(\alpha ,\beta ,\gamma )$ in the plane.

$\left(\frac{\alpha +0}{2},\frac{\beta +5}{2},\frac{\gamma +5}{2}\right)$ lies in the plane $x+y+z=7$

$\frac{\alpha +0}{2}+\frac{\beta +5}{2}+\frac{\gamma +5}{2}=7\Rightarrow \alpha +\beta +\gamma =4$

$\frac{\alpha -0}{1}=\frac{\beta -5}{1}=\frac{\gamma -5}{1}=\lambda$

 AQ is the line of reflection $Q=(-2,3,3)$

$\text{ direction ratios of reflected line are }-3,1\text{ and }-1\text{. Thus, its equation is }$

$\frac{x-1}{-3}=\frac{y-2}{1}=\frac{z-4}{-1}$