A function $y=f(x)$ has length on the interval $[0,7]$ given by the integral ${\displaystyle \underset{a}{\overset{b}{}}}\sqrt[\phantom{2}]{1+9co{s}^2(2x)}dx$ and passes through the point $(\frac{}{4},1).$

a$)$ Determine the correct values of a and $b.$

b$)$ Determine the equation for $y=f(x).$