Q.Write the axis to which the plane $by+cz+d=0$ is parallel.

Plane Parallel to a Coordinate Axis

The general equation of a plane is $Ax + By + Cz + D = 0$, whose normal vector is $\vec{n} = (A, B, C)$. A plane is parallel to a coordinate axis precisely when its normal is perpendicular to that axis, and this happens exactly when the coefficient of the corresponding variable is zero. So a missing variable in the equation is the tell-tale sign.

Concretely, if the $x$-term is absent, giving $By + Cz + D = 0$, the normal $(0, B, C)$ has no component along the $x$-direction $(1,0,0)$, so the plane runs parallel to the $x$-axis. Likewise $Ax + Cz + D = 0$ is parallel to the $y$-axis, and $Ax + By + D = 0$ is parallel to the $z$-axis. For instance, $2y + 3z = 6$ is parallel to the $x$-axis: no matter how $x$ changes, the equation is unaffected, so the plane extends infinitely in the $x$-direction. …