Conic Section Directrix Calculator – Parabola & Hyperbola
The directrix is a line used to define certain conic sections.
This tool focuses on axis-aligned (non-rotated) parabolas and hyperbolas,
so there is no \(xy\) term in the underlying equations.
1) Parabola: directrix from the standard form
A parabola is the set of points whose distance to a fixed point (the focus)
equals its distance to a fixed line (the directrix).
In standard axis-aligned form:
-
Vertical axis
\[
(x-h)^2 = 4p(y-k)
\]
Vertex \(V=(h,k)\), focus \(F=(h,k+p)\), and the directrix is:
\[
\boxed{y = k - p}
\]
-
Horizontal axis
\[
(y-k)^2 = 4p(x-h)
\]
Vertex \(V=(h,k)\), focus \(F=(h+p,k)\), and the directrix is:
\[
\boxed{x = h - p}
\]
The parameter \(p\) is signed. If \(p>0\), the parabola opens upward/right.
If \(p<0\), it opens downward/left.
2) Parabola: directrix from vertex form coefficient \(a\)
Many parabolas are given in vertex (function) form:
-
Vertical:
\[
y = a(x-h)^2 + k
\]
-
Horizontal:
\[
x = a(y-k)^2 + h
\]
Compare to the standard form by dividing:
\[
y-k=\frac{1}{4p}(x-h)^2
\quad\Longrightarrow\quad
a=\frac{1}{4p}
\quad\Longrightarrow\quad
\boxed{p=\frac{1}{4a}}
\]
(and similarly for the horizontal case). Once \(p\) is found, the directrix is still \(y=k-p\) or \(x=h-p\).
3) Hyperbola: directrices
A hyperbola has two directrices (one for each branch), and it also has an
eccentricity \(e>1\).
For axis-aligned hyperbolas centered at \((h,k)\):
-
Horizontal transverse axis
\[
\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1
\]
Define
\[
c=\sqrt{a^2+b^2},\qquad e=\frac{c}{a}.
\]
The directrices are vertical lines:
\[
\boxed{x = h \pm \frac{a}{e}}
\]
Using \(e=\frac{c}{a}\), this can be written as:
\[
\boxed{x = h \pm \frac{a^2}{c}}
\]
-
Vertical transverse axis
\[
\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1
\]
With the same definitions of \(c\) and \(e\), the directrices are horizontal lines:
\[
\boxed{y = k \pm \frac{a}{e}}
\quad\text{or}\quad
\boxed{y = k \pm \frac{a^2}{c}}.
\]
Hyperbola asymptotes (useful for sketching) are:
\[
y-k=\pm\frac{b}{a}(x-h)\quad\text{(horizontal case)},\qquad
y-k=\pm\frac{a}{b}(x-h)\quad\text{(vertical case)}.
\]
Example (requested parabola)
Given:
\[
y=\frac{x^2}{8}
\]
This matches \(y=a(x-h)^2+k\) with \(a=\frac{1}{8}\), \(h=0\), \(k=0\).
Compute:
\[
p=\frac{1}{4a}=\frac{1}{4\cdot\frac{1}{8}}=2.
\]
For a vertical parabola, the directrix is:
\[
\boxed{y=k-p=0-2=-2.}
\]
On the graph, the directrix line(s) are drawn as dashed lines and labeled with their equation(s).
