 Hi friends, I am Poodwa and today we will work out the following question. In the following, find the coordinates of the foot of the perpendicular drawn from the origin. 2x plus 3y plus 4z minus 12 is equal to 0. Let us begin with the solution now. Now we are given the equation of the plane is 2x plus 3y plus 4z minus 12 is equal to 0. Or we can write this as 2x plus 3y plus 4z is equal to 12. Let the coordinates of the foot of the perpendicular of point P from origin to the plane B x1, y1, z1. Then the direction ratios of line OPR, x1, y1, z1. Now writing the equation of the plane in normal form we have 2x upon under root 29 plus 3y upon under root 29 plus 4z upon under root 29 is equal to 12 upon under root 29. Where we have under root 29 is equal to under root of coefficient of x whole square. Now in this equation coefficient of x is 2. So we have 2 square plus coefficient of y whole square that is 3 square plus coefficient of z whole square that is 4 square. Here 2 upon under root 29, 3 upon under root 29 and 4 upon under root 29 are the direction cosines of OP. Now here this equation is of the form Lx plus My plus nz is equal to D which is the Cartesian equation of plane normal form and here L, M and N are the direction cosines of the normal and D is the distance of the plane from the origin. So here we get 2 upon under root 29, 3 upon under root 29 and 4 upon under root 29 are the direction cosines of OP. Now since the direction cosines and direction ratios of a line are proportional we have, now here x1, y1 and z1 are the direction ratios and 2 upon under root 29, 3 upon under root 29 and 4 upon under root 29 are the direction cosines. So we get x1 upon 2 upon under root 29 is equal to y1 upon 3 upon under root 29 is equal to z1 upon 4 upon under root 29 is equal to k. The k is some constant so we get that is x1 is equal to 2k upon under root 29, y1 is equal to 3k upon under root 29 and z1 is equal to 4k upon under root 29. Now substituting these values in the equation of the plane we get 2 into 2k upon under root 29 plus 3 into 3k upon under root 29 plus 4 into 4k upon under root 29 is equal to 12. This implies 4k plus 9k plus 16k is equal to 12 under root 29. This implies 29k is equal to 12 under root 29 which further implies k is equal to 12 upon under root 29. Now putting this value of k here in one we get thus putting the value of k in one we get x1 is equal to 2 upon under root 29 into 12 upon under root 29 which is equal to 24 upon 29. y1 is equal to 3 upon under root 29 into 12 upon under root 29 which is equal to 36 upon 29 and z1 is equal to 4 upon under root 29 into 12 upon under root 29 which is equal to 48 upon 29. Thus the coordinates of the foot of the perpendicular are 24 upon 29, 36 upon 29 and 48 upon 29. So we have got our answer as 24 upon 29, 36 upon 29, 48 upon 29. Hope you have understood the solution. Bye and take care.