JEE Main & Advanced Mathematics Three Dimensional Geometry Definition of Plane and Its Equations

If point \[P(x,y,z)\] moves according to certain rule, then it may lie in a 3-D region on a surface or on a line or it may simply be a point. Whatever we get, as the region of P after applying the rule, is called locus of P. Let us discuss about the plane or curved surface. If Q be any other point on it’s locus and all points of the straight line PQ lie on it, it is a plane. In other words if the straight line PQ, however small and in whatever direction it may be, lies completely on the locus, it is a plane, otherwise any curved surface.

(1) General equation of plane : Every equation of first degree of the form \[Ax+By+Cz+D=0\] represents the equation of a plane. The coefficients of \[x,\,\,y\] and z i.e., \[A,\,\,B,\,\,C\] are the direction ratios of the normal to the plane.

(2) Equation of co-ordinate planes : \[XOY\]-plane : \[z=\text{ }0,\] \[YOZ\]-plane : \[x=0,\] \[ZOX\]-plane : \[y=0\]

(3) Equation of plane in various forms :

(i) Intercept form : If the plane cuts the intercepts of length \[a,\,\,b,\,\,c\] on co-ordinate axes, then its equation is \[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\].

(ii) Normal form : Normal form of the equation of plane is \[lx+my+nz=p\], where \[l,\,\,m,\,\,c\] are the d.c.’s of the normal to the plane and \[p\] is the length of perpendicular from the origin.

(4) Equation of plane in particular cases : Equation of plane through the origin is given by \[Ax+By+Cz=0\].

i.e, if \[D=0\], then the plane passes through the origin.

(5) Equation of plane parallel to co-ordinate planes or perpendicular to co-ordinate axes

(i) Equation of plane parallel to \[YOZ\]-plane (or perpendicular to x-axis) and at a distance \['a'\] from it is \[x=a\].

(ii) Equation of plane parallel to \[ZOX\]-plane (or perpendicular to y-axis) and at a distance \['b'\] from it is \[y=b\].

(iii) Equation of plane parallel to \[XOY\]-plane (or perpendicular to z-axis) and at a distance \['c'\] from it is \[z=c\].

(6) Equation of plane perpendicular to co-ordinate planes or parallel to co-ordinate axes

(i) Equation of plane perpendicular to \[YOZ\]-plane or parallel to x-axis is \[By+Cz+D=0\].

(ii) Equation of plane perpendicular to \[ZOX\]-plane or parallel to y-axis is \[Ax+Cz+D=0\].

(iii) Equation of plane perpendicular to \[XOY\]-plane or parallel to z-axis is \[Ax+By+D=0\].

(7) Equation of plane parallel to a given plane : Plane parallel to a given plane \[ax+by+cz+d=0\] is \[ax+by+cz+{d}'=0\], i.e. only constant term is changed.

(8) Equation of plane passing through the intersection of two planes : Equation of plane through the intersection of two planes \[P={{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}=0\] and \[Q={{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}=0\] is \[P+\lambda Q=0\], where \[\lambda \] is the parameter.