If the objective function $z = 4x + 3y$ has maximum value on a line joining points $(3, a)$ and $(b, 2)$ where $a > 0, b > 0$ such that $a - b = 2$, then the maximum value of $z$ is:

54

The correct answer is Option (2) → 54

Given points: $(3,a)$ and $(b,2)$ with $a-b=2$

Objective function: $z = 4x + 3y$

Maximum occurs where slope of line joining points equals $-\frac{4}{3}$ (ratio of coefficients):

Slope: $m = \frac{2-a}{b-3} = -\frac{4}{3} \Rightarrow 2-a = -\frac{4}{3}(b-3) \Rightarrow 2-a = -\frac{4b-12}{3} \Rightarrow 4b - 3a = 6$

Also, $a-b=2 \Rightarrow a = b+2$

Substitute: $4b - 3(b+2) = 6 \Rightarrow 4b-3b-6=6 \Rightarrow b=12 \Rightarrow a = 14$

Compute $z$ at endpoints: $(3,14) \Rightarrow z = 4*3 + 3*14 = 54$

$(12,2) \Rightarrow z = 4*12 + 3*2 = 54$

Maximum value: $z_{\max} = 54$