= Hyperbolas with prescribed asymptotes
{title2=$L_1(x,y)L_2(x,y)=c$}
For two distinct intersecting <straight lines> through the origin, with independent real linear equations $L_1=0$ and $L_2=0$, the nondegenerate centred <hyperbolas> having exactly those <asymptotes> are
$$
L_1(x,y)L_2(x,y)=c,\qquad c\ne0.
$$
Indeed, the homogeneous <quadratic form> of a centred <hyperbola> vanishes in its two asymptotic directions and hence is a nonzero scalar multiple of $L_1L_2$. Its equation has no linear terms because the origin is its centre, and its constant term is nonzero by nondegeneracy. Conversely, the invertible <linear transformation> $(x,y)\mapsto(L_1,L_2)$ takes the displayed curve to $uv=c$, a <hyperbola>, and its ends approach $u=0$ or $v=0$. Distinct values of $c$ give disjoint <level sets>. In particular, the <rectangular hyperbolas> with <asymptotes> $y=\pm x$ are $x^2-y^2=c$, $c\ne0$.
Back to article page
