$\text{Area} = \frac{1}{2}ab$

$36 = \frac{1}{2}ab$

$b = \dfrac{72}{a}$

$\dfrac{x}{a} + \dfrac{y}{b} = 1$

At *C*(5, 2)

$\dfrac{5}{a} + \dfrac{2}{b} = 1$
$5b + 2a = ab$

$5 \left( \dfrac{72}{a} \right) + 2a = a\left( \dfrac{72}{a} \right)$

$\dfrac{360}{a} + 2a = 72$

$360 + 2a^2 = 72a$

$a^2 - 36a + 180 = 0$

$a = 30 ~ \text{ and } 6$

$b = 2.4 ~ \text{ and } ~ 12 \text{ respectively}$

For *a* = 30 and *b* = 2.4

$L_{AB} = \sqrt{30^2 + 2.4^2} = 30.10 ~ \text{units}$

For *a* = 6 and *b* = 12

$L_{AB} = \sqrt{6^2 + 12^2} = 13.42 ~ \text{units}$

Answer = [ C ]