Maths Paper 1, November 2024, question 5.2

The diagram below shows the graph of $f(x)=\frac{a}{x+p}+q$ with domain $(-\mathrm{\infty };2)\cup (2;\mathrm{\infty })$. The graph of $f$ cuts the $y$-axis at $(0;2)$. The straight line $g(x)=x-5$ is a line of symmetry of $f$. Determine the equation of the horizontal asymptote of $f$.