
Is this the correct solution something doesnot feel right


Is this the correct solution something doesnot feel right

Just like Blue said, you may better care how to write your thoughts on the paper.
Students like to use the $=\,$ sign for more kinds of things, while in math this has a rather strict meaning. So, instead rather use arrow or colon.. The other thing you messed up is $a$ alias $\sqrt{a^2}$ and $\sqrt{a}$. This latter is not needed now, fortunately.
So, once $a^2$ is found/defined to be $10$, it will stay $10$. Your last 3 lines with more exactness should go like this, for example:
(Gradients of the) Asymptotes
$a=\pm\sqrt{10}$, $\ b=\pm\sqrt{40/21}$
$\displaystyle\frac{b}{a}=\pm\sqrt{4/21}\simeq 0.43644$