In R³, let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P₁: x + 2y - z + 1 = 0 and P₂: 2x - y + z - 5 = 0. Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P₁. Which of the following points lie(s) on M?

Line L will be parallel to the line of intersection of P₁ and P₂. Let a, b and c be the direction ratios of line L → a + 2b - c = 0 and 2a - b + c = 0 → n₁ x → n₂ = ∣∣∣∣∣^i^j^k12-1-1∣∣∣∣∣ = ^i(2 - 1) ^j(-1 - 2) ^k(1 + 4) → a:b:c::1:-3:-5 Hence, the Equation of line L is x/1 = y/-3 = z/-5 as it passes through the origin. The foot of perpendicular from origin to plane P₁ is (-2, -1, 16) ∴Equation of projection of line L on plane P₁(M) is (x+2)/1 = (y+1)/-3 = (z-16)/-5 = c From the options given, only (-2, -1, 16) and (0, -4, -3) satisfy the equation of line of projection M