So I need to find the intersection between a hyperboloid ($z=\frac {y^2}{b^2}-\frac{x^2}{a^2}$) and some related plane ($bx+ay-z=0$).
I have tried solving for $z$ and equating the two: $$\frac{y^2}{b^2}-ay-\frac{x^2}{a^2}-bx=0$$
And then trying to find some parameter, which I tried $t=\frac xa$, however that leads to something messy: $$\frac{y^2}{b^2}-ay-t^2-abt=0$$ Any advice? I know the answers already, I just am having a lot of trouble deriving them.
As recommended, here is my solution for completing the squares. Once you have this form: $$\frac{y^2}{b^2}-ay-\frac{x^2}{a^2}-bx=0$$
You can group the $y$ terms and the $x$ terms. Substituting $u$ for $\frac{y}b$ and $v$ for $\frac{x}a$: $$(u^2-abu)-(v^2+abv)=0$$ $$[(u-\frac{ab}2)^2-\frac{a^2b^2}4]-[(v+\frac{ab}2)-\frac{a^2b^2}4]=0$$ $$(u-\frac{ab}2)^2-(v-\frac{ab}2)^2=0$$
Separate $v$ and $u$ to their own sides, and subsitute $y$ and $x$ back in: $$(\frac{y}b-\frac{ab}2)^2=(\frac{x}a-\frac{ab}2)^2$$
Declaring that your parameter is $t=\frac{x}a$, you end up with a form that is easily solvable for $y$ and $x$, and later $z$.