Short Answer 
The solution involves defining points on a parabola and calculating a height constraint for a trapezoid formed by these points. By setting up an equation and substituting known values, the value of ‘a’ is determined to be 1/12.
Step 1: Define Points and Parabola
Start by identifying key points on the x-axis. Let the points be J = (2, 0) and M = (4, 0), ensuring they lie on the x-axis and satisfy the relationship m < l < k. Next, express the coordinates of points L and K as functions of a. Since these points are located on a parabola, you can describe their y-coordinates using the equation y = a(x – 1)(x – 5).
Step 2: Apply Height Constraint
Calculate the height of the trapezoid formed by points L and K using the established height requirement of 2 units. The difference between the y-coordinates of points K and L must equal this height constraint. Set up the equation: a(k – 1)(k – 5) – a(l – 1)(l – 5) = 2 which will allow for the manipulation of the terms to eventually isolate a.
Step 3: Solve for a Using Known Values
With J and M defined, substitute their corresponding values into the equation to determine a. You have l = 2 and k = 4. Plugging these into the derived expression will yield: a = (4)^2 – (2)^2 – 10(4 – 2) + 4)/2. Following the simplification steps, you find that a = 1/12, confirming the intended result.