Finding Value of a in Functions

Consider two functions defined as follows:

Function $f(x) = x^2 + bx + c$ and function $g(x) = ax + k$, where $a$, $b$, $c$, and $k$ are constants. It is known that the two functions intersect at two distinct points, and the values of $f(1)$ and $g(1)$ are equal. Furthermore, it is given that the sum of the roots of $f(x) = g(x)$ is equal to $4$.

What is the value of $a$ in terms of $b$, $c$, and $k$?