 Hello and welcome to the session. The given question says, find the image of the point with coordinates 1, 2 and 3 in the plane x plus 2y plus 4z is equal to 38. Let's start with the solution and let q be the image of the point. Let us denote this point by p having coordinates 1, 2, 3 in the plane having equation x plus 2y plus 4z is equal to 38. Then pq is normal to the plane and therefore direction ratios of pq are 1, 2, 4. Now since pq passes through p having coordinates 1, 2 and 3 has direction ratios 1, 2, 4. Therefore, equation of pqs x minus 1 divided by 1 is equal to y minus 2 divided by 2 is equal to z minus 3 divided by 4 and let this be equal to r. So this implies that x is equal to r plus 1, y is equal to 2r plus 2 and z is equal to 4r plus 3. Now, let the coordinates the point q, r plus 1, 2r plus 2 and 4r plus 3. Also let r be the midpoint of pq then lies on the midpoint of pq. Therefore, coordinates of rr are r plus 1 plus 1 divided by 2 r plus 2 plus 2 divided by 2 and 3 plus 4r plus 3 plus 3 divided by 2. This is 4 equal to r plus 2 divided by 2. Here we have 2r plus 4 divided by 2 which gives r plus 2 and here we have 4r plus 6 divided by 2 which gives 2r plus 3. Now since this point r lies on the plane and the equation of the plane is x plus 2y plus 4z is equal to 38. Therefore, these coordinates must satisfy the equation of the line since lies on the plane having equation x plus 2y plus 4z is equal to 38. Therefore, we must have r plus 2 divided by 2 plus 2 times of r plus 2 plus 4 times of 2r plus 3 is equal to 38 which further implies that r plus 2 divided by 2 plus 2r plus 4 plus 8r plus 12 is equal to 38 or we have cross multiplying r plus 2 plus 4r plus 8 plus 16r plus 24 is equal to 76. Now this further implies that 21r is equal to 76 minus 34 which further implies that 21r is equal to 42 or r is equal to 2. Therefore, coordinates of q r plus 2 divided by 2, so 2 plus 2 divided by 2 gives 2. Then we have r plus 2 that is 4 and then we have 2r plus 3 that is 2 into 3 is 6 and 6 plus 3 gives 9. Hence the image of point 1, 2, 3 in the plane x plus 2y plus 4z is equal to 38 is 2, 4 and 9 which is a point q having coordinates 2, 4, 9. So, this completes the session by intake care.