A ray of light passing through the point A(1,2, 3), strikes the plane x + y + Z = 12 at B and on reflection passes
through point C(3,5, 9). The coordinates of point B are

$\text{ Let the image of point }A(1,2,3)\text{ about }x+y+z=12$$\text{ be }D(p,q,r)\text{. Then }$

$\frac{p-1}{1}=\frac{q-2}{1}=\frac{z-3}{1}=\frac{-2(1+2+3-12)}{1^2+1^2+1^2}$

$\therefore D(p,q,r)\equiv D(5,6,7)$

$\text{ Now equation of line }CD\text{ is }\frac{x-5}{3-5}=\frac{y-6}{5-6}=\frac{z-7}{9-7}\text{ or }$

$\frac{x-5}{-2}=\frac{y-6}{-1}=\frac{z-7}{2}=\lambda \text{ (say) }$

Point B is point of intersection of AC with the plane.

$\text{ Let }B\equiv (-2\lambda +5,-\lambda +6,2\lambda +7)$

This point lies on the plane. Therefore,

$\begin{array}{l}-2\lambda +5-\lambda +6+2\lambda +7=12\\ \lambda =6\\ B\equiv (-7,0,19)\end{array}$