Question

how do i find the vertices of a hyperbola

Answer 1

The vertices occur at the point closest to the centre of the hyperbola. So if you have something like y=1/x, then you need to minimise the distance function (i.e. minimise the distance between an arbitrary point on the hyperbola and the origin). So here, \[d^2=(x-0)^2+(y-0)^2=x^2+\frac{1}{x^2} \rightarrow 2dd'=2x-\frac{2}{x^3}\rightarrow d'=0 \iff x-\frac{1}{x^3}=0\]\[\rightarrow x^4-1=0 \rightarrow x = \pm 1\]are the only real roots. So your vertices occur at\[(1,1),(-1,-1)\]in this instance.