Rational Question for $a + b$ and Irrationality of $a^2 + b^2$

Question

I have looked into the question and need help. Find some $a,b$ ${\in}$ $\mathbb{R}$ such that $a + b$ ${\in}$ $\mathbb{Q}$, $a^2 + b^2 \not\in \mathbb{Q}$, and $\frac{a}{2} < b < a$. Or prove no such $a$ and $b$ exist.

Answer 1

How about $a = 10 + \pi $ and $b = 10 - \pi $?

Answer 2

let $a=r+ir$ and $b=r-ir$, $r$ is a rational number and $ir$ is irrational , $a^2+b^2$ will be irrational if $ir^2$ is irrational now if the question was other way around , i.e $a+b$ irrational $a^2+b^2$ cannot be rational , that's also wrong as we can take both of them as square roots of an integer