If the equation k x y+10 x+8 y +16=0 represents a pair of lines, then

Step-by-step solution

The condition for a second-degree equation ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 to represent a pair of lines is abc + 2fgh - af^2 - bg^2 - ch^2 = 0 . Comparing kxy + 10x + 8y + 16 = 0 with the general form yields: a = 0 , b = 0 , h = k 2 , g = 5 , f = 4 , c = 16 Substituting into the condition: (0)(0)(16) + 2(4)(5) ( k 2 ) - (0)(4^2) - (0)(5^2) - 16 ( k 2 )^2 = 0 20k - 4k^2 = 0 4k(5 - k) = 0 The solutions are k = 0 and k = 5 , both representing degenerate pairs of lines. The final answer is C