A ray of light passing through the point `A(1,2,3)` , strikews the plane `x y+z=12a tB` and on reflection passes through point `C(3,5,9)dot` Find the coordinate so point `Bdot`

Explanation

Let the image of point `A(1,2, 3)` about x+y+z=12 be `D(p, q, r)`. Then
`" " (p-1)/(1)=(q-2)/(1)=(z-3)/(1)= (-2(1+2+3-12))/(1^(2)+1^(2)+1^(2))`
`therefore" "D(p, q, r)-=D(5, 6, 7)`
Now equation of line CD is `(x-5)/(3-5)=(y-6)/(5-6)=(z-7)/(9-7) or (x-5)/(-2)=(y-6)/(-1)=(z-7)/(2)=lamda` (say)
Point B is point of intersection of AC with the plane.
Let `" "B-=(-2lamda+5, -lamda+6, 2lamda+7)`
This point lies on the plane. Therefore,
`" "-2lamda+5-lamda+6+2lamda+7=12`
or `" "lamda=6`
`therefore" "B-=(-7, 0, 19)`