Numerical Methods: Euler to Runge–Kutta · Challenge problem

The second-order Runge–Kutta family $k_1 = f(t_n, y_n)$, $k_2 = f(t_n + ch,\ y_n + chk_1)$, $y_{n+1} = y_n + h\left[\left(1 - \dfrac{1}{2c}\right)k_1 + \dfrac{1}{2c}k_2\right]$ has one free parameter $c > 0$. For the pure-quadrature equations $y' = g(t)$, one step integrates $g$ over $[t_n, t_n + h]$. For which $c$ is the step exact for every quadratic polynomial $g$? Express your answer as a common fraction.