Let the plane a x+b y+c z+d=0 bisect the line joining the points (4,-3,1) and (2,3,-5) at the right angles. If a, b, c, d are integers, then the minimum value of (a^2+b^2+c^2+d^2 ) is

Step-by-step solution

Step-1 Using mid point concept: Given that the equation of the plane is a x+b y+c z+d=0 and the line passed the plane having the coordinates as P(4,-3,1) and Q(2,3,-5) . Let M(x, y, z) be the mid-point of the line passing through the plane. Using the mid-point formula find the coordinate of M(x, y, z) . x= x₁+x₂ 2 aligned & x= 4+2 2 & x= 6 2 & x=3 aligned And aligned y & = y₁+y₂ 2 y & = -3+3 2 y & = 0 2 y & =0 aligned And aligned z & = z₁+z₂ 2 z & = 1-5 2 z & = -4 2 z & =-2 aligned Therefore, the mid-point M(x, y,