Directrix of a Parabola: Definition, Equation, Examples
Directrix of a Parabola: Definition, Equation, Examples
TL;DR
The directrix of a parabola is a fixed line, perpendicular to the axis, such that every point on the curve is exactly as far from the directrix as it is from the focus. This article covers the equidistance definition, the directrix equation for all four standard forms, how to find and derive it, six worked examples, and common mistakes.
What Is the Directrix of a Parabola?
The directrix of a parabola is the fixed straight line such that every point on the parabola is the same distance from the directrix as it is from the focus. It is always perpendicular to the axis of symmetry, it never touches the curve, and it sits on the opposite side of the vertex from the focus.
This is the defining rule of the parabola itself. If F is the focus, P is any point on the curve, and M is the foot of the perpendicular from P to the directrix, then:
PF = PM for every point P on the parabola.
That single equation, "distance to the focus equals perpendicular distance to the directrix," is what makes the shape a parabola rather than any other curve. The focus and the directrix are partners: the focus of a parabola is the point side of the definition, and the directrix is the line side. The vertex is the one point caught exactly in the middle, so it lies halfway between the focus and the directrix.
How Do You Find the Directrix of a Parabola?
The directrix comes straight from the standard equation, and it is always the mirror image of the focus across the vertex. For a parabola with vertex at the origin opening to the right:
y² = 4ax ⇒ directrix x = -a,
where a is the distance from the vertex to the focus, and also the distance from the vertex to the directrix. The two distances are equal by definition, which is exactly why the directrix sits at -a when the focus sits at +a. The four standard orientations follow the same single number a:
| Equation | Opens | Focus | Directrix |
|---|---|---|---|
| y² = 4ax | right | (a,0) | x = -a |
| y² = -4ax | left | (-a,0) | x = a |
| x² = 4ay | up | (0,a) | y = -a |
| x² = -4ay | down | (0,-a) | y = a |
The recipe is the same every time. Read 4a off the coefficient, divide by 4 to get a, then place the directrix a units from the vertex on the opposite side from the focus, as a line perpendicular to the axis.
Why the directrix lands at x = -a
This is worth deriving once, because then the equation stops being a thing to memorize. Take the simplest case: vertex at the origin, focus at the unknown point (a,0), and a vertical directrix at the unknown line x=-a. A point (x,y) on the parabola must obey "distance to focus equals perpendicular distance to directrix." The distance to the focus is ( \sqrt{(x - a)^2 + y^2} ), and the perpendicular distance to the vertical line x = -a is simply x + a:
( \sqrt{(x - a)^2 + y^2} = x + a. )
Square both sides and expand:
x² - 2ax + a² + y² = x² + 2ax + a².
The x² and a² terms cancel, leaving y² = 4ax, the standard form. So setting the directrix at x=-a is what produces the equation y² = 4ax, the directrix position is not a separate fact, it falls out of the equidistance rule directly.
Finding the Directrix When You Are Given the Equation in Another Form
Real problems rarely hand you a clean y² = 4ax. Two situations come up constantly, so it helps to name them before the examples.
From a quadratic y=ax²+bx+c: Complete the square to get it into vertex form (x-h)²=4p(y-k), read the vertex (h,k) and the value of p, then the directrix is y=k-p.
From a focus and a vertex: The directrix is the reflection of the focus across the vertex. If the vertex is (0,0) and the focus is (0,3), the directrix is the line y=-3, the same distance on the far side.
Both reduce to the same idea: locate the vertex, find a, then step a units the other way to lay down a line perpendicular to the axis.
Examples of the Directrix of a Parabola
Example 1 - Find the directrix of the parabola y²=12x
Compare with y²=4ax: here 4a=12, so a=3. The parabola opens right with focus at (3,0), so the directrix sits 3 units to the left of the vertex, perpendicular to the axis.
Final answer: the directrix is x=-3.
Example 2 - Find the directrix of the parabola x²=-8y
Wrong attempt. A student reads 4a=8, gets a=2, and writes the directrix as y=-2, taking the number positive and putting the line below the vertex out of habit. Check this against the equation: the right side is -8y, so the parabola opens downwards, which means the focus sits below the vertex. The directrix must therefore sit above it.
Correct. Match with x²=-4ay, so 4a=8 and a=2. The parabola opens down, focus at (0,-2), so the directrix is the line on the opposite side: y=2.
Final answer: the directrix is y=2.
Example 3 - Find the directrix of y²=-20x
Match with y²=-4ax: 4a=20, so a=5, and the parabola opens left with focus at (-5,0). The directrix sits to the right of the vertex:
directrix x=5.
Final answer: the directrix is x=5.
Example 4 - A parabola has the equation (y-1)²=8(x-2)
The vertex has shifted to (2,1). From 4a=8, a=2, and the parabola opens right. The directrix is vertical, a units to the left of the new vertex:
directrix x=h-a=2-2=0.
Final answer: the directrix is x=0 (the y-axis).
Example 5 - Find the directrix of y=\frac{1}{8}x²
Rewrite into conic form. Multiply through: 8y=x², so x²=8y. Match with x²=4ay, giving 4a=8 and a=2. The parabola opens up, focus at (0,2), so the directrix is below the vertex:
directrix y=-2.
Final answer: the directrix is y=-2.
Example 6 - A parabola opens upward, has vertex at (3,-1), and a focus at (3,1). Find its directrix
The directrix is the reflection of the focus across the vertex. The focus is 2 units above the vertex, so a=2, so the directrix is the horizontal line 2 units below the vertex:
directrix y=-1-2=-3.
Final answer: the directrix is y=-3.
Why the Directrix of a Parabola Matters
- It is what defines the curve. Drop the directrix and "parabola" loses its meaning.
- Conic sections as one family. Each conic has a focus and a directrix, and the ratio of the two distances, the eccentricity, is what separates them.
- Antenna and microphone design. Parabolic microphones rely on the same focus-directrix geometry to pick a single distant sound out of background noise.
Key Takeaways
- The directrix of a parabola is the fixed line where every point on the curve is equidistant from the directrix and the focus.
- The focus and directrix are mirror images across the vertex.
- The most common mistake is putting the directrix on the same side as the focus.