For two positive numbers $a , b$, a continuous random variable $X$ has a range of $0 \leqq X \leqq a$, and the graph of the probability density function is as shown. When $\mathrm { P } \left( 0 \leqq X \leqq \frac { a } { 2 } \right) = \frac { b } { 2 }$, find the value of $a ^ { 2 } + 4 b ^ { 2 }$. [4 points]