Question

Find the equation of the plane that passes through $x-$ axis and point $(3,2,4)$.

Answer 1

Equation of plane passing through point $(3,2,4)a(x-3)+b(y-2)+c(z-4)=\mathrm{0...}\left(1\right)$
$\because$ Plane is perpendicular to $X-$axis
$\therefore a=0,d=0\Rightarrow by+cz=\mathrm{0...}\left(2\right)$
From $\left(1\right),a=0$ so
$b(y-2)+c(z-4)=\mathrm{0...}\left(3\right)$
$\Rightarrow by-2b+cz-4c=0$
$\Rightarrow by+cz-2b-4c0$
$\Rightarrow -2b=c$
$[\because by+cz=0from(2\left)\right]$
$\Rightarrow b=0-2c$
$\therefore$ Equation of plane passing through point $(3,2,4)$ and $X-$axis $b(y-2)+c(z-4)=0$
$\Rightarrow -2c(y-2)+c(z-4)=0$
$\Rightarrow -2y+4+z-4=0$
$\Rightarrow 2y-z=0$