Question

Find the image of the point $P(3,5,7)$ in the plane $2x+y+z=16$

Answer 1

Let the image of point $P$ be $(x,y,z)$

$\therefore$ we know that,

$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}=\frac{-2(a{x}_1+b{y}_1+c)}{a^2+b^2+c^2}$

$\therefore \frac{x-3}{2}=\frac{y-5}{1}=\frac{z-7}{1}=\frac{-2(6+5+7-16)}{4+1+1}$

$\therefore \frac{x-3}{2}=\frac{y-5}{1}=\frac{z-7}{1}=\frac{-2\left(2\right)}{6}$

$\therefore \frac{x-3}{2}=\frac{-4}{6};y-5=\frac{-4}{6};z-7=\frac{-4}{6}$

$\therefore x=3-\frac{4}{3}=\frac{5}{3}=\frac{5}{3}$

$y=5-\frac{4}{6}=\frac{26}{6}=\frac{13}{3}$

$z=7-\frac{4}{6}=\frac{38}{6}=\frac{19}{3}$

$\therefore Imagein(\frac{5}{3},\frac{13}{3},\frac{19}{3})$