If for a >0, the feet of perpendicular from the points A(a,-2a,3) and B(0,4,5) on the plane lx+ my + nz = 0 are points C(0, -a, -1) and D respectively, then the length of line segment CD is equal to :

Direction ratios of the line AC are ⟨-a,a,-4⟩Hence the equation of the plane which is passing through the origin and having normal  direction ratios ⟨-a,a,-4⟩ is ax-ay+4z=0 The point C(0,-a,-1) lies on the above plane, hence a2-4=0⇒a=2 Equation of the plane is 2x-2y+4z=0⇒x-y+2z=0 The projection of B(0,4,5) on the above plane is h1=k−4−1=l−52=−66⇒h=−1,k=5,l=3 Hence, D(-1,5,3) and C(0,-2,-1) Therefore, the distance between C and D is 1+49+16=66