Mathematics Extension 1

Consider the hyperbola $y=\dfrac{1}{x}$ and the circle $(x-c)^2+y^2=c^2$, where $c$ is a constant.

(i) Show that the $x$-coordinates of any points of intersection of the hyperbola and circle are zeros of the polynomial $P(x)=x^4-2cx^3+1$.

Consider the hyperbola $y=\dfrac{1}{x}$ and the circle $(x-c)^2+y^2=c^2$, where $c$ is a constant.

(ii) The graphs of $y=x^4-2cx^3+1$ for $c=0.8$ and are shown.

By considering the given graphs, or otherwise, find the exact value of
$c>0$ such that the hyperbola $y=\frac{1}{x}$ and the circle
$(x-c{)}^{}$ intersect at only one point.