Asymptotes of a hyperbola

Question

Is this the correct solution something doesnot feel right

Answer 1

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

$a=\pm\sqrt{10}$, $\ b=\pm\sqrt{40/21}$

$\displaystyle\frac{b}{a}=\pm\sqrt{4/21}\simeq 0.43644$