Find The Directrix Calculator

Easily calculate the equation of the directrix, the focus, and the standard form of a parabola using our directrix calculator.

Parabola Directrix Calculator

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What is a Directrix?

In geometry, a parabola is defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). The directrix is a line perpendicular to the axis of symmetry of the parabola. Every point on the parabola is the same distance from the focus as it is from the directrix. This directrix calculator helps you find the equation of this line.

Understanding the directrix is crucial when working with parabolas, whether in algebra, physics (like the path of projectiles under gravity or the shape of parabolic reflectors), or engineering. The relationship between the focus and the directrix determines the shape and orientation of the parabola. The distance from the vertex to the focus is the same as the distance from the vertex to the directrix, and this distance is called the focal length (|p|). Our directrix calculator uses this principle.

Common misconceptions include thinking the directrix passes through the parabola or the focus. In reality, the directrix never intersects the parabola and is on the opposite side of the vertex from the focus.

Directrix Formula and Mathematical Explanation

The equation of the directrix depends on whether the parabola opens vertically or horizontally, and the position of its vertex (h, k) and its focal length 'p'.

For a parabola with vertex at (h, k):

• If the parabola opens vertically (up or down), its equation is (x – h)² = 4p(y – k). The axis of symmetry is x = h, and the directrix is a horizontal line: y = k – p. The focus is at (h, k + p).
• If the parabola opens horizontally (left or right), its equation is (y – k)² = 4p(x – h). The axis of symmetry is y = k, and the directrix is a vertical line: x = h – p. The focus is at (h + p, k).

The value 'p' is the directed distance from the vertex to the focus. If p > 0, a vertical parabola opens upwards, and a horizontal parabola opens to the right. If p < 0, a vertical parabola opens downwards, and a horizontal parabola opens to the left. The directrix calculator above implements these formulas.

Variables in Parabola Equations

Variable	Meaning	Unit	Typical Range
h	x-coordinate of the vertex	Length units	Any real number
k	y-coordinate of the vertex	Length units	Any real number
p	Focal length (directed distance from vertex to focus)	Length units	Any non-zero real number
x, y	Coordinates of any point on the parabola	Length units	Varies

Table explaining the variables used in the directrix and parabola formulas.

Practical Examples (Real-World Use Cases)

Example 1: Vertical Parabola

Suppose you have a parabolic satellite dish. Its vertex is at (0, 0), and it opens upwards with a focal length (p) of 2 units. We want to find its directrix.

• Vertex (h, k) = (0, 0)
• Focal length (p) = 2
• Orientation: Vertical (opens up since p > 0)

Using the formula for a vertical parabola, the directrix is y = k – p = 0 – 2 = -2. So, the directrix is the line y = -2. The focus would be at (0, 0 + 2) = (0, 2). The directrix calculator would give y = -2.

Example 2: Horizontal Parabola

Consider a parabola with its vertex at (1, 3) and a focal length 'p' of -4, opening horizontally.

• Vertex (h, k) = (1, 3)
• Focal length (p) = -4
• Orientation: Horizontal (opens left since p < 0)

For a horizontal parabola, the directrix is x = h – p = 1 – (-4) = 1 + 4 = 5. So, the directrix is the line x = 5. The focus would be at (1 + (-4), 3) = (-3, 3). Using the directrix calculator with these inputs confirms x = 5.

How to Use This Directrix Calculator

1. Select Orientation: Choose whether the parabola opens vertically or horizontally using the radio buttons.
2. Enter Vertex Coordinates: Input the h (x-coordinate) and k (y-coordinate) of the parabola's vertex.
3. Enter Focal Length (p): Input the value of 'p'. Remember 'p' can be positive or negative, but not zero. 'p' is the distance from the vertex to the focus.
4. Calculate: Click the "Calculate" button (or the results will update automatically as you type).
5. Read Results: The calculator will display the equation of the directrix as the primary result, along with the coordinates of the focus, the value of 4p, and the standard equation of the parabola. A simple graph will also be shown.
6. Reset/Copy: Use the "Reset" button to clear inputs to default values or "Copy Results" to copy the output.

The directrix calculator provides immediate feedback, allowing you to quickly find the directrix and related parameters.

Key Factors That Affect Directrix Results

1. Vertex Coordinates (h, k): The position of the vertex directly influences the 'k' or 'h' value in the directrix equation (y = k – p or x = h – p). Changing the vertex shifts the entire parabola and its directrix.
2. Focal Length (p – magnitude): The absolute value of 'p' determines the distance between the vertex and the directrix (and vertex and focus). A larger |p| means the directrix is further from the vertex, and the parabola is wider.
3. Focal Length (p – sign): The sign of 'p' determines the direction of opening and thus where the directrix is relative to the vertex. For vertical parabolas, p > 0 means it opens up and the directrix is below the vertex; p < 0 means it opens down and the directrix is above. For horizontal, p > 0 is right, directrix left; p < 0 is left, directrix right.
4. Orientation (Vertical/Horizontal): This fundamentally changes the form of the directrix equation from y = constant to x = constant and which coordinate of the vertex is used with 'p'.
5. Value of 4p: While not directly the directrix, 4p is the coefficient in the standard equation and is derived from 'p', which is used for the directrix.
6. Accuracy of Inputs: Small errors in h, k, or p will lead to inaccuracies in the calculated directrix. Ensure precise input values.

Understanding these factors helps in interpreting the results from the directrix calculator and understanding the geometry of the parabola.

Frequently Asked Questions (FAQ)

What is a parabola?

A parabola is a U-shaped curve where any point is at an equal distance from a fixed point (the focus) and a fixed line (the directrix).

What is the focus of a parabola?

The focus is a fixed point inside the parabola used in its definition. For a vertical parabola (x-h)²=4p(y-k), it's at (h, k+p); for horizontal (y-k)²=4p(x-h), it's at (h+p, k). The directrix calculator also shows the focus.

Can the focal length 'p' be zero?

No, if 'p' were zero, the equation would not represent a parabola (it would degenerate to a line or point), and the directrix would be undefined in the context of the distance definition used here. Our directrix calculator requires p ≠ 0.

How do I find the directrix if the parabola equation is not in standard form?

You first need to complete the square to convert the equation to one of the standard forms: (x – h)² = 4p(y – k) or (y – k)² = 4p(x – h). Once in standard form, you can identify h, k, and p, and then use the formulas y = k – p or x = h – p, or use the directrix calculator with h, k, and p.

What is the axis of symmetry?

It's a line that divides the parabola into two mirror images. For a vertical parabola, it's x=h; for a horizontal one, it's y=k.

Is the directrix always a line?

Yes, for a standard parabola, the directrix is always a straight line.

Where is the directrix located relative to the vertex and focus?

The vertex is always exactly midway between the focus and the directrix. The directrix is on the opposite side of the vertex from the focus.

Can I use this directrix calculator for rotated parabolas?

No, this directrix calculator is designed for parabolas that open vertically or horizontally (axis of symmetry parallel to the y-axis or x-axis). Rotated parabolas have more complex equations and directrix forms.