Why Dish Antennas Are Shaped Like Parabolas, Not Bowls of Any Curve

A parabola is the set of all points in a plane that are equally far from a fixed point, the focus, and a fixed line, the directrix.

That single rule gives the curve its useful shape: signals arriving straight at a parabolic dish all bounce to the focus, which is where the receiver sits.

This part covers deriving the equation, the four standard forms, the latus rectum and sketching, and writing parabolas from given conditions.

**Take the focus at $S(a,0)$ and the directrix $x=-a$; a point $P(x,y)$ is on the parabola when its distance from $S$ equals its distance from the directrix, which simplifies to $y^2=4ax$.

Derivation.**

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

The vertex is the origin, exactly midway between the focus and the directrix.

Worked check. For $a=3$, the point $(3,6)$ lies on $y^2=12x$, since $36=36$.

$$\text{distance to focus }(3,0)=6\text{distance to directrix }x=-3\text{ is }3+3=6$$

An everyday example. The stream of water from a garden hose follows a parabolic path through the air.

The substance. Every point on the curve satisfies the equal-distance rule, not just the vertex — which is why the equation holds for all of them.

**With vertex at the origin, $y^2=4ax$ opens right, $y^2=-4ax$ opens left, $x^2=4ay$ opens up, and $x^2=-4ay$ opens down, for $a>0$.

- $y^2=4ax$** — axis $y=0$; focus $(a,0)$; directrix $x=-a$; opens right
- **$y^2=-4ax$** — axis $y=0$; focus $(-a,0)$; directrix $x=a$; opens left
- **$x^2=4ay$** — axis $x=0$; focus $(0,a)$; directrix $y=-a$; opens up
- **$x^2=-4ay$** — axis $x=0$; focus $(0,-a)$; directrix $y=a$; opens down

Worked example 1. $y^2=12x$: $4a=12$, $a=3$. Focus $(3,0)$, directrix $x=-3$, axis the x-axis.

Worked example 2. $x^2=-16y$: $4a=16$, $a=4$. Focus $(0,-4)$, directrix $y=4$, opens downward.

Worked example 3. $y^2=-8x$: $a=2$. Focus $(-2,0)$, directrix $x=2$, opens left.

An everyday example. The arc of water from a fountain in a park opens downward, like $x^2=-4ay$.

The substance. The squared variable tells you the axis, and the sign on the other side tells you which way the curve opens.

**The latus rectum is the chord through the focus perpendicular to the axis; for $y^2=4ax$ its ends are $(a,2a)$ and $(a,-2a)$, so its length is $4a$.

Why.** Put $x=a$ in $y^2=4ax$: $y^2=4a^2$, so $y=\pm 2a$.

Worked example 1. $y^2=12x$ has $a=3$:

$$\text{latus rectum}=12\text{ends }(3,6)\text{ and }(3,-6)$$

Worked example 2. $x^2=-16y$ has $a=4$:

$$\text{latus rectum}=16\text{ends }(8,-4)\text{ and }(-8,-4)$$

Sketching steps:

- Mark the vertex at the origin and the focus
- Draw the directrix as a dashed line
- Plot the ends of the latus rectum
- Draw a smooth curve through the vertex and both ends, symmetric about the axis

An everyday example. A dish antenna on a rooftop has its receiver held on arms at the focus, and the width of the dish at that height is the latus rectum.

The substance. **A larger $a$ gives a wider, flatter parabola**, since the latus rectum $4a$ grows.

**Use the given information to decide the orientation and the value of $a$, then write the matching standard form; if only a point is given, the orientation must also be stated.

Worked example 1 — focus and directrix.** Focus $(6,0)$, directrix $x=-6$:

$$a=6y^2=24x$$

Worked example 2 — vertex and focus. Vertex $(0,0)$, focus $(-2,0)$: the parabola opens left with $a=2$:

$$y^2=-8x$$

Worked example 3 — axis and a point. Vertex at the origin, axis along the y-axis, passing through $(2,3)$. Use $x^2=4ay$:

$$4=4a\times 3a=\frac{1}{3}{x}^2=\frac{4}{3}y\text{or}3x^2=4y$$

Worked example 4 — same point, other axis. Axis along the x-axis through $(2,3)$. Use $y^2=4ax$:

$$9=8a{y}^2=\frac{9}{2}x$$

Worked example 5 — latus rectum. Ends of the latus rectum at $(3,6)$ and $(3,-6)$ give $a=3$ and $y^2=12x$.

An everyday example. An engineer designing a headlight reflector with the bulb $3$ cm from the vertex uses $y^2=12x$ for its cross-section.

The substance. The same point lies on two different standard parabolas, which is why the axis must be known.