Finding functions such that the maximum is in certain values

Let $a_1, a_2, b_1, b_2, r_1, r_2 \geq 0$ with $a_1 > b_1$ and $b_2 > a_2$ be pre-determined constants. Define the function $h$ as $$h(x, y) := f(x, y) (a_1 r_1 + a_2 r_2) + g(x, y) (b_1 r_1 + b_2 r_2) - x - y.$$I now want to find functions $f$ and $g$ such that the maximum of $h$ is achieved for $(x, y) = (r_1, r_2)$.

My idea was that this would mean that$$ a_1 f(x,y) + b_1 g(x, y) = \ln(x)$$$$ a_2 f(x, y) + b_2 g(x, y) = ln(y)$$so that $h(x,y) = ln(x) r_1 + ln(y) r_2 - x-y$.However, I'm unsure about how to derive $f$ and $g$ and for what values of $a_1, a_2, b_1, b_2 \geq 0$ there exist such functions. Any help about how to approach such a question would be highly appreciated.