Parabola

A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). Its eccentricity is $e = 1$.

Parabolas are everywhere in physics: projectiles fly along them, satellite dishes have parabolic cross-sections, suspension-bridge cables hang in approximately parabolic curves.

Standard form (right-opening)

Place the focus at $F(a, 0)$ and the directrix at the line $x = -a$ (where $a > 0$). A point $P(x, y)$ on the parabola satisfies $|PF| =$ distance from $P$ to directrix, i.e. $\sqrt{(x - a)^2 + y^2} = |x + a|.$ Squaring: $(x - a)^2 + y^2 = (x + a)^2$, which simplifies to $\boxed{y^2 = 4 a x.}$

This is the standard form of a right-opening parabola.

Key features.

• Vertex: $(0, 0)$.
• Focus: $(a, 0)$.
• Directrix: $x = -a$.
• Axis: $x$-axis.
• Latus rectum (chord through focus perpendicular to axis): length $4a$, endpoints $(a, \pm 2a)$.

The four standard parabolas

Here $a > 0$ in each.

Shifted parabolas

If the vertex is at $(h, k)$ and the axis is parallel to $x$-axis: $(y - k)^2 = 4 a (x - h)$.

Reading the vertex from $(y + 2)^2 = 8(x - 3)$: vertex $(3, -2)$, axis horizontal, $4 a = 8 \Rightarrow a = 2$, focus $(3 + 2, -2) = (5, -2)$, directrix $x = 3 - 2 = 1$.

Worked examples

Example 1. Find focus, directrix, axis, vertex, and latus rectum of $y^2 = 12 x$.

$4 a = 12 \Rightarrow a = 3$. Vertex: $(0, 0)$. Focus: $(3, 0)$. Directrix: $x = -3$. Axis: $x$-axis. Latus rectum length: $12$.

Example 2. Find the equation of the parabola with focus $(0, -2)$ and directrix $y = 2$.

Vertex midway: $(0, 0)$. Axis: $y$-axis. Opens downward (focus below). $a = 2$. Equation: $x^2 = -4 \cdot 2 \cdot y = -8 y$.

Example 3. Find the equation of a parabola with vertex $(1, 2)$, focus $(4, 2)$.

Axis horizontal (focus and vertex have same $y$). Opens right. $a = 4 - 1 = 3$. Equation: $(y - 2)^2 = 12(x - 1)$.

Example 4. Find the length of the latus rectum of $y^2 = -20 x$.

$4 a = 20 \Rightarrow a = 5$. Latus rectum: $4 a = 20$.

Example 5. A parabola passes through $(2, 3)$ with vertex at origin and axis along $x$-axis. Find its equation.

$y^2 = 4 a x \Rightarrow 9 = 8 a \Rightarrow a = 9/8$. Equation: $y^2 = (9/2) x$.

Try it yourself

1. Find focus and directrix of $y^2 = 16 x$.
2. Find focus and directrix of $x^2 = 8 y$.
3. Find focus and directrix of $y^2 = -10 x$.
4. Equation of a parabola with focus $(3, 0)$ and directrix $x = -3$.
5. Equation of a parabola with vertex at origin, axis along $y$-axis, passing through $(4, 8)$.
6. Length of the latus rectum of $x^2 = -16 y$.
7. Find the vertex of $(y - 1)^2 = 4(x + 2)$.
8. A parabola has focus $(2, 5)$ and directrix $y = -1$. Find its equation.
9. Sketch and identify $y^2 = 8 x$.
10. Find the equation of the parabola with focus $(0, 4)$ and vertex $(0, 0)$.
11. The latus rectum of a parabola is $8$ and the axis is $y$-axis with vertex at origin. Find the equation (two cases).
12. A parabola with vertex $(0, 0)$ passes through $(3, -6)$ and has axis along the $x$-axis. Find its equation.

Pitfalls / Tricks

• The "$4a$" in $y^2 = 4 a x$ is not the latus rectum coefficient , it equals the latus rectum length.
• Always identify the axis from which variable is squared: $y^2 = \dots$ means horizontal axis; $x^2 = \dots$ means vertical axis.
• Sign of $a$ (or coefficient of the linear term) determines direction of opening.
• Insight. The focus is the "magic point" , every ray parallel to the axis, when it hits the parabola, reflects through the focus. This is why satellite dishes work.