Hyperbola ($x^2/a^2-y^2/b^2=1$)

A hyperbola is a conic section with two unbounded branches. In suitable coordinates its equation is $x^2/a^2-y^2/b^2=1$ with $a,b>0$. The unit hyperbola is parametrized by $x=\cosh t$, $y=\sinh t$ on its right branch, relating it to the hyperbolic functions.

For example, $x^2/9-y^2/16=1$ has foci of a conic at $(\pm 5,0)$, because $5^2=3^2+4^2$. Its focal-distance difference has absolute value $6$, while its eccentricity of a conic is $5/3>1$. At the point $(3,0)$, the focal distances are $8$ and $2$, verifying that difference. The asymptotes are $y=\pm 4x/3$; a distant point approaches those directions while remaining on one of the two branches. The focal-distance difference, rather than a constant sum, distinguishes this hyperbola from an ellipse.

For the two centred hyperbolas $x^2-y^2=c$ and $4x^2-y^2=d$, with nonzero real $c,d$, eliminating their quadratic terms givesThus they meet in four real points if both quantities are positive, two if one is zero and the other positive, and none if either is negative. Both quantities cannot vanish when $c,d\ne 0$. The choices $(c,d)=(1,5),(1,4),(1,1)$ respectively realize these three counts. This reduction to a linear system in $x^2,y^2$ is useful for intersections of axis-aligned conic sections; the independent sign choices must be counted after solving that linear system.

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 areIndeed, the homogeneous quadratic form of a centred hyperbola vanishes in its two asymptotic directions and hence is a nonzero scalar multiple of $L_1{L}_2$. 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 $x^2-y^2=c$, $c\ne 0$.

A hyperbola whose two asymptotes are perpendicular. After a suitable choice of coordinates it has equation $xy=c$ for a nonzero constant $c$. In the form $x^2/a^2-y^2/b^2=1$, it is rectangular exactly when $a=b$.

For nonzero real $a,b$, the identity ${\cosh }^{2}t-{\sinh }^{2}t=1$ eliminates the parameter. The additional condition $x/a=\cosh t\ge 1$ is essential: the parametrization traces only the branch with the sign of $a$. Every point of that branch is obtained by $t=\mathrm{arsinh}(y/b)$. Its vertex is $(a,0)$ and its asymptotes are $y=\pm (b/a)x$. An algebraic equation alone would also include the opposite branch, which this real parametrization never reaches.

For $x^2/a^2-y^2/b^2=1$, let $c=\sqrt{a^2+b^2}$ and place the foci of a conic at $(\pm c,0)$. On the right branch write $x=a\cosh t$, $y=b\sinh t$. Direct expansion givesSince $c>a>0$, both square roots are positive and $d_{-}-d_{+}=2a$. On the reflected left branch the signed difference reverses, while its absolute value stays $2a$. This proves the constant-difference characterization of a hyperbola and keeps track of which branch a hyperbolic function parametrization covers.