How the Focus and Directrix Shape Every Parabola

Slice a double cone with a flat plane and the edge of the cut can be a circle, an ellipse, a parabola or a hyperbola. These conic sections describe the paths of thrown balls, planets and comets, and the shapes of dish antennas and cooling towers.

This part covers conics as sections of a cone, the four standard parabolas, their key features, and equations from a focus and directrix.

A conic section is the curve cut from a double cone by a plane, and every conic other than the circle is the path of a point whose distance from a fixed point, the focus, is a constant multiple e of its distance from a fixed line, the directrix; the latus rectum is the chord through the focus perpendicular to the axis.

Types of section. If the plane makes angle $\beta$ with the axis of a cone whose semi-vertical angle is $\alpha$:

- $\beta =90°$ — circle
- $\alpha <\beta <90°$ — ellipse
- $\beta =\alpha$ — parabola
- $\beta <\alpha$ — hyperbola

Eccentricity. $e=\frac{\text{distance from focus}}{\text{distance from directrix}}$: $e=1$ for a parabola, $e<1$ for an ellipse and $e>1$ for a hyperbola.

Worked example. A point on a conic is 6 cm from the focus and 4 cm from the directrix. Then $e=\frac{6}{4}=1.5>1$, so the conic is a hyperbola.

Degenerate cases. When the plane passes through the vertex of the cone, the section is a point, a line or a pair of lines.

An everyday example. The patch of light a torch throws on a wall is a circle when aimed straight, an ellipse when tilted, and a parabola or hyperbola when tilted further.

The substance. A circle has no directrix — it is the limiting case $e=0$.

**Taking the focus at $(a,0)$ and the directrix $x=-a$ gives $y^2=4ax$, and turning the picture gives the other three standard forms, $y^2=-4ax$, $x^2=4ay$ and $x^2=-4ay$, each with its vertex at the origin.

Derivation.** For $P(x,y)$, the distance to the focus equals the distance to the directrix:

$$\sqrt{(x-a{)}^2+y^2}=x+a\Rightarrow x^2-2ax+a^2+y^2=x^2+2ax+a^2\Rightarrow y^2=4ax$$

The four forms with $a>0$:

- $y^2=4ax$ — opens to the right
- $y^2=-4ax$ — opens to the left
- $x^2=4ay$ — opens upward
- $x^2=-4ay$ — opens downward

**Sketching $y^2=8x$.** Here $4a=8$, so $a=2$. Plot $(0,0)$, $(2,4)$, $(2,-4)$, $(8,8)$ and $(8,-8)$; check $(8,8)$: $64=8\times 8$.

An everyday example. The main cables of a suspension bridge over a river, carrying a uniform deck, hang in a parabola that opens upward, like $x^2=4ay$.

The substance. The unsquared variable names the axis of symmetry — in $y^2=4ax$ the curve is symmetric about the x-axis, because y and $-y$ give the same x.

**For $y^2=4ax$ the vertex is the origin, the focus is $(a,0)$, the directrix is $x=-a$, the axis is the x-axis and the latus rectum has length $4a$, and the other standard forms follow the same pattern with x and y or the signs exchanged.

Features for $a>0$:**

- $y^2=4ax$ — focus $(a,0)$, directrix $x=-a$
- $y^2=-4ax$ — focus $(-a,0)$, directrix $x=a$
- $x^2=4ay$ — focus $(0,a)$, directrix $y=-a$
- $x^2=-4ay$ — focus $(0,-a)$, directrix $y=a$
- Latus rectum of length $4a$ in every case

Worked example. For $x^2=-12y$: $4a=12$, so $a=3$.

- Opens downward, with focus $(0,-3)$
- Directrix $y=3$; the axis is the y-axis
- Latus rectum 12, with end points $(6,-3)$ and $(-6,-3)$

Check $(6,-3)$: $36=-12\times (-3)$.

An everyday example. **A torch reflector shaped like $y^2=4x$, in centimetres,** needs its bulb at the focus $(1,0)$, 1 cm from the vertex, to throw a parallel beam.

The substance. The focus always lies inside the curve and the directrix outside it — a quick check on every sign.

**For any focus and directrix, set the distance of $P(x,y)$ from the focus equal to its perpendicular distance from the directrix and simplify; applied problems place the vertex at the origin, read one point from the data and solve for a.

Worked example.** Find the parabola with focus $(1,1)$ and directrix $x+y=0$.

$$\sqrt{(x-1{)}^2+(y-1{)}^2}=\frac{|x+y|}{\sqrt{2}}$$

Squaring gives $2(x^2-2x+1+y^2-2y+1)=x^2+2xy+y^2$, so

$$x^2-2xy+y^2-4x-4y+4=0$$

Check: the vertex lies midway between the focus and the directrix, at $\left(\frac{1}{2},\frac{1}{2}\right)$, and $0.25-0.5+0.25-2-2+4=0$.

Worked example 2. A dish antenna is 40 cm across and 10 cm deep. With the vertex at the origin and $x^2=4ay$, the rim point $(20,10)$ gives $400=40a$, so $a=10$ and the receiver sits 10 cm above the vertex.

An everyday example. A parabolic arch over a school gate, 8 m wide and 4 m high, fits $x^2=-4ay$ through $(4,-4)$, so $a=1$ and the arch is $x^2=-4y$.

The substance. A tilted directrix produces an xy term — the standard forms apply only when the axis is parallel to a coordinate axis.