Parametric parabola and hyperbola points

Conic sections (circle, ellipse, parabola, hyperbola) are the analytic curves used in CAD. Their parametric forms with centre or vertex at the origin are:

• Parabola $y^2=4ax$: $x=at^2$, $y=2at$, $-\infty<t<\infty$.
• Hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$: $x=a\sec\theta$, $y=b\tan\theta$ (or $x=a\cosh t$, $y=b\sinh t$).

Parabola $y^2=12x$

$4a=12\Rightarrow a=3$, so $x=3t^2$, $y=6t$.

Check at $t=2$: $y^2=144=12\times12$.

Tangent slope: $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{6}{6t}=\dfrac1t$. At $t=1$ the slope is 1 (45 deg).

Hyperbola with $a=3$, $b=4$

$x=3\sec\theta$, $y=4\tan\theta$.

Check at 45 deg: $18/9-16/16=2-1=1$. The parameter must stay away from $\theta=\pm90^\circ$, where $\sec\theta$ is infinite; the range $-90^\circ<\theta<90^\circ$ gives the right branch only.

Answer: parabola points (0,0), (0.75,3), (3,6), (12,12), slope at t = 1 is 1; hyperbola points (3,0), (3.464,2.309), (4.243,4.0).