If the distance of the point P (43, , ), <0 , from the line r =4 i - k + (2 i +3 k ), R along a line with direction ratios 3,-1,0 is 13 10 , then ²+ ² is equal to _ _ _ _

Step-by-step solution

Point P(43, α, β) has distance 13 10 from line r = 4 i - k + (2 i + 3 k ) . The line passes through A(4, 0, -1) with direction d = (2, 0, 3) . Using distance formula: d = | AP d | | d | . With AP = (39, α, β+1) , the cross product gives AP d = (3α, 2β - 115, -2α) . Thus | AP d | = 13α^2 + (2β - 115)^2 and | d | = 13 . Setting 13α^2 + (2β - 115)^2 13 = 13 10 gives α^2 + (2β - 115)^2 13 = 1690 . Solving with the constraint β < 0: α = 5, β = -13, hence α^2 + β^2 = 25 + 169 = 170 .