Find the values of the constant c for which the line 4y 2x

Find the values of the constant c for which the line 4y = 2x + c is a tangent to the curve y = 4x + 8/x.

Solution

If the line 4y = 2x + c or, y = x/2 + c/4 is a tangent to the curve y = 4x + 8/x, then their intersection must be a unique point. At the pointof intersection, we have ( on equating the two values of y) 4x + 8/x = x/2 + c/4 or, 4x – x/2 – c/4 + 8/x = 0 or, (7/2 )x - c/4 + 8/x = 0 or, (7/2)x² - cx/4 + 8 = 0 or, 14x² - cx + 32 = 0 ….(1). Then, x = [ c ± { (-c)² – 4 (14)(32)}]/ 2 * 28 . Since the quadratic equation (1) must have a unique solution, the discriminant must be zero. Thus, { (-c)² – 4 (14)(32)} = 0 or, c² – 1792 = 0 or, c² = 1792, so that c = ± 167