Find the equation of the hyperbola, if it passes through $(1;0)$ and has the asymptotes $x=0$ and $y=1$.

Question

I know I have to rotate it 45 degrees I just don't know how. I don't even know how this rotation thing works. I would really appreciate any help and if you can help me understand the rotation thing I promise to pray for you and your family every night. Thanks in advance!

P.S.Please do not bully me if I'm asking stupid stuff.I am a sensitive person.

Answer 1

A rotated, translated equilateral hyperbola has equation $$y=\frac{ax+b}{cx+d}$$ horizontal asymptote is $y=\frac{a}{c}$, vertical is $x=-\frac{d}{c}$.

Here we have $$ \begin{cases} \frac{a}{c}=1&a=c\\ -\frac{d}{c}=0&d=0\\ \frac{a+b}{c+d}=0&a+b=0\to b=-a\\ \end{cases} $$ So the equation is $$y=\frac{ax-a}{ax}\to y=\frac{x-1}{x}$$

Answer 2

The equation of the asymptotes of $xy=c^2$ are $xy=0$

So, the equation of the hyperbola will be $$x(y-1)=d$$

As it passes through $(1,0);$ $$d=1\cdot(0-1)$$