Focal Properties, Latus Rectum and Parametric Form of a Hyperbola

1. Distance between directrices. Exactly as for the ellipse: $d(dd')=\left|\dfrac{a}{e}-\left(-\dfrac{a}{e}\right)\right|=\dfrac{2a}{e}$.

2. End points and length of the latus rectum. Let $L\equiv(ae,l)$ (upper end point through the right focus). Since $L$ is on the hyperbola: $\dfrac{(ae)^2}{a^2}-\dfrac{l^2}{b^2}=1 \Rightarrow e^2-\dfrac{l^2}{b^2}=1 \Rightarrow \dfrac{l^2}{b^2}=e^2-1=\dfrac{b^2}{a^2}$ (using $b^2=a^2(e^2-1)$), giving $l^2=\dfrac{b^4}{a^2} \Rightarrow l=\dfrac{b^2}{a}$. So $L\equiv\left(ae,\dfrac{b^2}{a}\right)$, $L'\equiv\left(ae,-\dfrac{b^2}{a}\right)$, and the full latus rectum has length $l(LL')=\dfrac{2b^2}{a}$ — the same formula as the ellipse's.

3. Difference of focal distances is constant. For $P$ on the hyperbola, $SP=e\cdot PM$ and $S'P=e\cdot PM'$. Subtracting:

So the DIFFERENCE of the focal distances of any point on the hyperbola is the constant $2a$ — the length of the transverse axis. This is the hyperbola's analogue of the ellipse's constant-SUM property, and is likewise the basis of an alternative "difference of distances" definition of the hyperbola.

4. Auxiliary circle and parametric form. The circle with the transverse axis $AA'$ as diameter, $x^2+y^2=a^2$, is the auxiliary circle. Let $P(x,y)$ be a point on the hyperbola; draw $PM\perp OX$ and let the tangent from $M$ touch the auxiliary circle at $Q$, with $\angle XOQ=\theta$, so $Q=(a\cos\theta,a\sin\theta)$. Using the right-triangle relation $x=OM=OQ\sec\theta=a\sec\theta$ (since $OQ=a$ and $\angle QOM=\theta$), we get $x=a\sec\theta$. Substituting $P(a\sec\theta,y)$ into the hyperbola's equation:

$\dfrac{a^2\sec^2\theta}{a^2}-\dfrac{y^2}{b^2}=1 \;\Longrightarrow\; \dfrac{y^2}{b^2}=\sec^2\theta-1=\tan^2\theta \;\Longrightarrow\; y=\pm b\tan\theta.$ …