Homographic hyperbola from its two foci

Text

Write the implicit and explicit equations of a homographic hyperbola with one focus $F_1(0;0)$ and the other at $F_2(4;4)$.

Solution

The centre is the midpoint of the foci: $C(2;2)$. The focal axis has slope $1$ (consistent with a homographic hyperbola with horizontal/vertical asymptotes). The focal half-distance is $c=\tfrac12\,\overline{F_1F_2}=\tfrac12\sqrt{16+16}=2\sqrt2.$ For a rectangular hyperbola $c=a\sqrt2$, so $a=2$ and, from $a=\sqrt{2k}$, we get $k=2>0$ (foci on the rising bisector). Implicit form: $(x-2)(y-2)=2.$ Explicit form: $y=2+\dfrac{2}{x-2}=\dfrac{2x-2}{x-2}$.

$\boxed{(x-2)(y-2)=2\qquad\Longleftrightarrow\qquad y=\frac{2x-2}{x-2}}$