 Hello and welcome to the session. In this session we are going to discuss equations of some particular planes. The equation AX plus BY plus CZ plus K is equal to 0 represents a system of planes parallel to the plane AX plus BY plus CZ plus D is equal to 0. So, K being a parameter, let the given plane be AX plus BY plus CZ plus D is equal to 0. Mark the equation as 1. Now again let A1X plus B1Y plus C1Z plus D1 is equal to 0. Mark the equation as 2. Now planes 1 and 2 are parallel if 1 upon A is equal to B1 upon B is equal to C1 upon C is equal to say lambda. Therefore A1 is equal to A lambda, B1 is equal to B lambda, C1 is equal to C lambda. Now substituting the values of A1, B1 and C1 in equation 2 we get A lambda X plus B lambda Y plus C lambda Z plus D1 is equal to 0. All we can also write it as AX plus BY plus CZ plus D1 upon lambda is equal to 0 which can also be written as AX plus BY plus CZ plus K is equal to 0 where K is equal to D1 upon lambda equation of the plane parallel to the given plane plus BY plus CZ plus D is equal to 0 is given by AX plus BY plus CZ plus K is equal to 0. Next we have the equation plus BY plus CZ plus K is equal to 0 represents a system of planes perpendicular to the line with direction ratios. A, B, C the equation A1X plus B1Y plus C1Z plus D1 plus K into A2X plus D2Y plus C2Z plus D2 is equal to 0 represents a system of the planes passing through the line of intersection of the planes 1X plus B1Y plus C1Z plus D1 is equal to 0 and A2X plus B2Y plus C2Z plus D2 is equal to 0. K is being the parameter now we shall discuss equations of the planes parallel to the coordinate planes. The coordinate planes are XY plane, YZ plane and XZ plane. Now X is equal to A represents a plane parallel to the YZ plane at a distance A from it similarly Y is equal to B and Z is equal to C are planes parallel to the XZ plane and XY plane respectively. Now we shall discuss equations of planes perpendicular to the coordinate planes is plus BY plus CZ plus D is equal to 0 is perpendicular to the plane X is equal to 0 then A into 1 plus B into 0 plus C into 0 is equal to 0 that is A is equal to 0. Therefore BY plus CZ plus D is equal to 0 is perpendicular to the plane is equal to 0 similarly AX plus CZ plus D is equal to 0 is perpendicular to the plane Y is equal to 0 AX plus BY plus D is perpendicular to the plane Z is equal to 0. Next we have equations of the planes perpendicular to the coordinate axis. Now any plane perpendicular to X axis is evidently parallel to the YZ plane and its equation is equal to A similarly the equations of the planes perpendicular to the Y axis Y is equal to B and perpendicular to the Z axis is Z is equal to C. Now we shall discuss equations of the planes parallel to the coordinate axis. The equation BY plus CZ plus D is equal to 0 is parallel to the X axis the equation AX plus CZ plus D is equal to 0 is parallel to Y axis and the equation AX plus BY plus D is equal to 0 is parallel to Z axis. Now the two straight lines given by X minus X1 by N1 is equal to Y minus Y1 by M1 is equal to Z minus Z1 by N1 and X minus X2 by L2 is equal to Y minus Y2 by M2 is equal to Z minus Z2 by M2 our co-planar if, if the determinant containing elements X2 minus X1 L1 L2 by 2 minus Y1 M1 M2 Z2 minus Z1 N1 M2 is equal to 0 in general we can write it as the determinant containing elements X minus X1 L1 L2 by minus Y1 M1 M2 Z minus Z1 M1 M2 is equal to 0 and this is the equation of the plane containing both the lines in case the lines are co-planar the condition shows that the point with the coordinates X2, Y2, Z2 lies on the first plane and the point with the coordinates X1, Y1, Z1 lies on the second these two planes are identical and contain both the intersecting lines this completes our session hope you enjoyed this session