how many positive integer solutions to the following equation? $a^2 + b^2 + 25 = ab + 5a + 5b$
I have tried looking for a factorisation that could solve this question but couldn't find anything useful - found $(a+b+5)^2$ - don't know if this is useful
The equation does look similar to an equation of a circle - can you use this idea?

Let $x = a+b \to x^2 - 2ab + 25 = ab + 5x \to x^2 -5x + 25 = 3ab \leq 3\cdot \dfrac{(a+b)^2}{4} = \dfrac{3x^2}{4} \to 4x^2 - 20x + 100 \leq 3x^2 \to x^2 - 20x + 100 \leq 0 \to (x-10)^2 \leq 0 \to x = 10, a = b = \dfrac{10}{2} = 5$.
