Directrix of a Parabola — Definition, Formula & Examples

Directrix of a Parabola

A line perpendicular to the axis of symmetry used in the definition of a parabola. A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.

Parabola with focus point, directrix line below, and equal distances L1 (to focus) and L2 (to directrix) shown by arrows.

Example:

Graph of upward-opening parabola with vertex at (3,-2), focus above vertex, vertical axis of symmetry at x=3, and horizontal...

This is a graph of the parabola with all its major features labeled: axis of symmetry, focus, vertex, and directrix.

Key Formula

\begin{gathered}\text{For a parabola with vertex } (h, k) \text{ opening upward: } y = k - p\\\text{where the equation of the parabola is } (x - h)^2 = 4p(y - k)\end{gathered}

Where:

$h$ = The x-coordinate of the vertex
$k$ = The y-coordinate of the vertex
$p$ = The directed distance from the vertex to the focus (positive means the parabola opens upward or rightward; negative means it opens downward or leftward)
$y = k - p$ = The equation of the directrix for a vertical parabola
$x = h - p$ = The equation of the directrix for a horizontal parabola of the form (y - k)² = 4p(x - h)

Worked Example

Problem: Find the directrix of the parabola given by the equation x² = 12y.

Step 1: Identify the standard form. The equation x² = 12y matches the form (x − h)² = 4p(y − k) with vertex at (0, 0).

(x - 0)^2 = 4p(y - 0)

Step 2: Solve for p by setting 4p equal to the coefficient of y.

4p = 12 \implies p = 3

Step 3: Since the vertex is at (0, 0) and the parabola opens upward (p > 0), the focus is at (0, 3) and the directrix is the horizontal line below the vertex.

y = k - p = 0 - 3 = -3

Step 4: Verify: pick a point on the parabola, say (6, 3). Distance to the focus (0, 3) is 6. Distance to the directrix y = −3 is |3 − (−3)| = 6. The distances are equal, confirming the answer.

\sqrt{(6-0)^2 + (3-3)^2} = 6 \quad\text{and}\quad |3 - (-3)| = 6 \;\checkmark

Answer: The directrix is the line y = −3.

Another Example

This example involves a horizontal parabola opening to the left (negative p), showing that the directrix can be a vertical line rather than a horizontal one.

Problem: Find the directrix of the parabola y² + 8x = 0.

Step 1: Rewrite the equation in standard form by isolating the squared term.

y^2 = -8x

Step 2: This matches the horizontal form (y − k)² = 4p(x − h) with vertex (0, 0). Set 4p equal to −8.

4p = -8 \implies p = -2

Step 3: Since p is negative, the parabola opens to the left. The focus is at (−2, 0). For a horizontal parabola, the directrix is a vertical line given by x = h − p.

x = 0 - (-2) = 2

Step 4: Verify: the point (−2, 4) satisfies y² = −8x because 16 = −8(−2) = 16. Distance to focus (−2, 0) is 4. Distance to the line x = 2 is |−2 − 2| = 4. Equal, as expected.

\sqrt{(-2-(-2))^2 + (4-0)^2} = 4 \quad\text{and}\quad |-2 - 2| = 4 \;\checkmark

Answer: The directrix is the vertical line x = 2.

Frequently Asked Questions

What is the difference between the focus and the directrix of a parabola?

The focus is a fixed point and the directrix is a fixed line. Together they define the parabola: every point on the curve is the same distance from the focus as it is from the directrix. The focus sits inside the curve, while the directrix lies outside, on the opposite side of the vertex.

How do you find the directrix from a quadratic equation in the form y = ax² + bx + c?

First, rewrite the equation in vertex form y = a(x − h)² + k. Then note that 4p = 1/a (since a = 1/(4p)), so p = 1/(4a). The directrix is the line y = k − p = k − 1/(4a). For example, if y = 2x², then p = 1/8 and the directrix is y = −1/8.

Is the directrix always below the parabola?

No. The directrix is always on the opposite side of the vertex from the focus. For an upward-opening parabola the directrix is below, but for a downward-opening parabola the directrix is above. For horizontal parabolas, the directrix is a vertical line to the left or right of the vertex.

Directrix vs. Focus

	Directrix	Focus
Type of object	A line	A point
Position relative to vertex	On the opposite side from where the parabola opens, at distance p from the vertex	Inside the parabola, at distance p from the vertex
Formula (vertical parabola, vertex (h, k))	y = k − p	(h, k + p)
Role in definition	Every point on the parabola is equidistant from this line and the focus	Every point on the parabola is equidistant from this point and the directrix

Why It Matters

You encounter the directrix when studying conic sections in precalculus and analytic geometry courses. It is essential for deriving the equation of a parabola from scratch and for solving problems involving satellite dishes, headlight reflectors, and other applications where the reflective property of parabolas matters. Understanding the directrix also prepares you for the directrix definitions of ellipses and hyperbolas, which use a similar distance-ratio approach.

Common Mistakes

Mistake: Placing the directrix on the same side of the vertex as the focus.

Correction: The directrix is always on the opposite side of the vertex from the focus. If the focus is above the vertex, the directrix is below it (and vice versa). Remember: the vertex sits exactly halfway between the focus and the directrix.

Mistake: Confusing the sign of p when computing the directrix from y = ax².

Correction: When a > 0 the parabola opens upward, so p = 1/(4a) is positive and the directrix y = −p is below the vertex. When a < 0, p is negative and the directrix y = −p is above the vertex. Always check that the directrix ends up on the side opposite the opening.

Related Terms

Focus of a Parabola — The fixed point paired with the directrix
Parabola — The curve defined by the focus and directrix
Axis of Symmetry of a Parabola — Line perpendicular to the directrix through the focus
Vertex of a Parabola — Midpoint between the focus and directrix
Locus — Set of points satisfying the distance condition
Distance from a Point to a Line — Used to measure distance to the directrix
Directrices of an Ellipse — Analogous concept for ellipses
Directrices of a Hyperbola — Analogous concept for hyperbolas