Finding Directrix And Focus Calculator

Easily find the focus, directrix, vertex, and axis of symmetry of a parabola with our Directrix and Focus Calculator.

Parabola Calculator

What is a Directrix and Focus Calculator?

A directrix and focus calculator is a tool used to determine the key elements of a parabola given certain parameters. A parabola is a curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). Our directrix and focus calculator helps you find the coordinates of the focus, the equation of the directrix, the coordinates of the vertex, and the equation of the axis of symmetry based on the vertex (h, k) and the focal length (p), along with the parabola's orientation.

This calculator is useful for students studying conic sections in algebra and geometry, engineers, and anyone working with parabolic shapes, such as satellite dishes or reflector telescopes. The directrix and focus calculator simplifies the process of analyzing these curves.

Common misconceptions include thinking the focus is always inside the "cup" of the parabola (which is true) or that the directrix passes through the vertex (it does not; the vertex is midway between the focus and directrix).

Directrix and Focus Formula and Mathematical Explanation

The standard equations for a parabola with vertex at (h, k) are:

1. If the parabola opens vertically (up or down): (x – h)² = 4p(y – k)
2. If the parabola opens horizontally (left or right): (y – k)² = 4p(x – h)

Where:

• (h, k) are the coordinates of the vertex.
• p is the focal length – the directed distance from the vertex to the focus. If p > 0, the parabola opens towards the positive axis direction (up or right); if p < 0, it opens towards the negative axis direction (down or left).

Using these equations, the directrix and focus calculator derives:

For a parabola opening up or down ((x – h)² = 4p(y – k)):

• Vertex: (h, k)
• Focus: (h, k + p)
• Directrix: y = k – p
• Axis of Symmetry: x = h

For a parabola opening left or right ((y – k)² = 4p(x – h)):

• Vertex: (h, k)
• Focus: (h + p, k)
• Directrix: x = h – p
• Axis of Symmetry: y = k

The directrix and focus calculator applies these formulas based on the selected orientation.

Variables Table

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

Table explaining the variables used in the directrix and focus calculations.

Practical Examples

Let's see how the directrix and focus calculator works with some examples.

Example 1: Parabola opening upwards

• Vertex (h, k) = (2, 3)
• Focal length p = 2
• Orientation: Opens Up/Down

Using the directrix and focus calculator (or formulas):

• Vertex: (2, 3)
• Focus: (2, 3 + 2) = (2, 5)
• Directrix: y = 3 – 2 = 1
• Axis of Symmetry: x = 2
• Equation: (x – 2)² = 4 * 2 * (y – 3) => (x – 2)² = 8(y – 3)

Example 2: Parabola opening to the left

• Vertex (h, k) = (-1, 1)
• Focal length p = -3
• Orientation: Opens Left/Right

Using the directrix and focus calculator:

• Vertex: (-1, 1)
• Focus: (-1 + (-3), 1) = (-4, 1)
• Directrix: x = -1 – (-3) = 2
• Axis of Symmetry: y = 1
• Equation: (y – 1)² = 4 * (-3) * (x – (-1)) => (y – 1)² = -12(x + 1)

How to Use This Directrix and Focus Calculator

1. Enter Vertex Coordinates: Input the values for 'h' (x-coordinate) and 'k' (y-coordinate) of the parabola's vertex.
2. Enter Focal Length: Input the value for 'p', the focal length. Remember, p cannot be zero. A positive 'p' means the parabola opens up (for vertical) or right (for horizontal), and a negative 'p' means it opens down or left.
3. Select Orientation: Choose whether the parabola opens "Up/Down" (equation form (x-h)²=4p(y-k)) or "Left/Right" (equation form (y-k)²=4p(x-h)) from the dropdown menu.
4. View Results: The directrix and focus calculator will instantly display the focus coordinates, directrix equation, vertex coordinates, axis of symmetry, and the parabola's equation.
5. See the Graph: A visual representation of the parabola with its focus, directrix, and vertex will be drawn on the canvas.
6. Reset: Click the "Reset" button to clear the inputs and set them back to default values.
7. Copy Results: Click "Copy Results" to copy the calculated values to your clipboard.

The results from the directrix and focus calculator give you a complete picture of the parabola's key features.

Key Factors That Affect Parabola Elements

Several factors influence the position and shape of the parabola, as calculated by the directrix and focus calculator:

• Vertex (h, k): This directly sets the base position of the parabola. Changes in h or k shift the entire parabola, including its focus and directrix, horizontally or vertically.
• Focal Length (p): This determines the "width" or "openness" of the parabola and the distance between the vertex and both the focus and directrix. A smaller |p| value means a narrower parabola, while a larger |p| value means a wider parabola.
• Sign of p: The sign of 'p' dictates the direction the parabola opens. For a vertical parabola, p > 0 opens up, p < 0 opens down. For a horizontal parabola, p > 0 opens right, p < 0 opens left.
• Orientation: Whether the parabola's axis of symmetry is vertical (opens up/down) or horizontal (opens left/right) fundamentally changes the standard equation and the relative positions of the focus and directrix.
• Value of 4p: This term, appearing in the standard equations, is the "latus rectum" length, which is the width of the parabola through the focus, perpendicular to the axis of symmetry.
• Coordinate System: The values of h, k, and p are relative to the chosen Cartesian coordinate system.

Understanding these factors helps in interpreting the output of the directrix and focus calculator and how the parabola is formed.

Frequently Asked Questions (FAQ)

Q: What is a parabola? A: A parabola is a U-shaped curve defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Q: Can the focal length 'p' be zero? A: No, 'p' cannot be zero. If p=0, the equation degenerates and does not form a parabola. The directrix and focus calculator will show an error if p is zero.

Q: What happens if 'p' is negative? A: If 'p' is negative, the parabola opens in the negative direction along its axis of symmetry (downwards for a vertical parabola, leftwards for a horizontal one). The focus will be 'p' units away from the vertex in that negative direction.

Q: How is the vertex related to the focus and directrix? A: The vertex is the midpoint between the focus and the point on the directrix that lies on the axis of symmetry. It's the point on the parabola closest to the directrix and the focus.

Q: What is the axis of symmetry? A: The axis of symmetry is a line that divides the parabola into two mirror images. It passes through the vertex and the focus.

Q: Can I use this directrix and focus calculator for parabolas not centered at the origin? A: Yes, the calculator uses the vertex (h, k), so it works for parabolas centered at any point (h, k), not just the origin (0, 0).

Q: What are real-world applications of parabolas? A: Parabolic shapes are used in satellite dishes, car headlights, reflector telescopes, and the paths of projectiles under gravity (ignoring air resistance).

Q: Does the directrix ever intersect the parabola? A: No, the directrix never touches or intersects the parabola.

Related Tools and Internal Resources

• Quadratic Equation Solver: Useful for solving equations related to the intersection of lines and parabolas.
• Distance Calculator: Calculate the distance between two points, like the vertex and focus.
• Midpoint Calculator: Find the midpoint between two points, relevant to the vertex's position relative to focus and directrix intersection with axis of symmetry.
• Equation of a Line Calculator: Useful for understanding the directrix equation.
• Circle Equation Calculator: Explore another conic section.
• Ellipse Calculator: Learn about ellipses, another type of conic section related to parabolas.