 Hi friends I am Pooja and today we will work out the following question. Find the equation of the plane through the intersection of the planes 3x-y plus 2z-4 is equal to 0 and x plus y plus z minus 2 is equal to 0 and the point 2, 2, 1. Now the equation of the plane that passes through the intersection of two given planes a1x plus b1y plus c1y z plus d1 is equal to 0 and a2x plus b2y plus c2z plus d2 is equal to 0 is a1x plus b1y plus c1z plus d1 plus lambda into a2x plus b2y plus c2z plus d2 is equal to 0. So this is the key idea behind our question. Let us begin with the solution now. Now we are given the two planes as 3x minus y plus 2z minus 4 is equal to 0 and x plus y plus z minus 2 is equal to 0. Now the equation of the plane that passes through the intersection of the two given planes is given by, now by key idea we know that the equation of the plane that passes through the intersection of two given planes is given by this equation. So here the equation of the plane that passes through the intersection of these two planes is given by 3x minus y plus 2z minus 4 plus lambda into x plus y plus z minus 2 is equal to 0 where we have lambda is some real number. We mark this as equation 1. Now since the given plane passes through the point 2 comma 2 comma 1 therefore it must satisfy equation 1 that is putting x is equal to 2, y is equal to 2 and z is equal to 1 in equation 1 we get 3 into 2 minus 2 plus 2 into 1 minus 4 plus lambda into 2 plus 2 plus 1 minus 2 is equal to 0. This implies 6 minus 2 plus 2 minus 4 plus lambda into 3 is equal to 0 and this further implies 2 plus 3 lambda is equal to 0 which implies lambda is equal to minus 2 upon 3. Now putting lambda is equal to minus 2 upon 3 in equation 1 we get 3x minus y plus 2z minus 4 minus 2 upon 3 into x plus y plus z minus 2 is equal to 0. Now taking LCM and solving this equation further we get this implies 9x minus 3y plus 6z minus 12 minus 2x minus 2y minus 2z plus 4 is equal to 0 and this implies 7x minus 5y plus 4z minus 8 is equal to 0. Thus the equation of the plane that passes through the intersection of these two planes and this point is 7x minus 5y plus 4z minus 8 is equal to 0. This is our answer. Hope you have understood the solution. Bye and take care.