Find The Parabola With Focus And Directrix Calculator

Enter the coordinates of the focus and the equation of the directrix to find the equation of the parabola and its properties. Our find the parabola with focus and directrix calculator makes it simple.

Property	Value
Focus	(2, 3)
Directrix	y = 1
Vertex
p
Axis of Symmetry
Equation

Summary of the parabola’s properties.

What is a Find the Parabola with Focus and Directrix Calculator?

A “find the parabola with focus and directrix calculator” is a tool used to determine the equation of a parabola when you know the coordinates of its focus (a fixed point) and the equation of its directrix (a fixed line). A parabola is defined as the set of all points that are equidistant from the focus and the directrix. This calculator helps you find the standard equation, vertex, axis of symmetry, and the value ‘p’ (distance from vertex to focus/directrix) for the parabola.

This calculator is useful for students studying conic sections in algebra or analytic geometry, engineers, physicists, and anyone working with parabolic shapes, such as reflectors or antennas. It automates the calculations involved in deriving the parabola’s equation from its fundamental components: the focus and directrix. Common misconceptions include thinking the focus is part of the parabola (it’s not) or that the directrix touches the parabola (it doesn’t).

Find the Parabola with Focus and Directrix Formula and Mathematical Explanation

A parabola is defined by a focus point (fx, fy) and a directrix line. There are two main cases:

1. Directrix is horizontal (y = k): The parabola opens upwards or downwards. Its axis of symmetry is vertical (x = fx). The vertex (h, v) is midway between the focus and the directrix, so h = fx and v = (fy + k) / 2. The distance from the vertex to the focus (and to the directrix) is p = |fy – k| / 2, with the sign of ‘p’ depending on whether the focus is above or below the directrix (p = (fy – k)/2). The standard equation is:
   
   (x – h)² = 4p(y – v)
   
   Substituting h, v, and p: (x – fx)² = 2(fy – k)(y – (fy + k)/2)
2. Directrix is vertical (x = k): The parabola opens sideways (left or right). Its axis of symmetry is horizontal (y = fy). The vertex (h, v) is h = (fx + k) / 2 and v = fy. The distance p = (fx – k) / 2. The standard equation is:
   
   (y – v)² = 4p(x – h)
   
   Substituting h, v, and p: (y – fy)² = 2(fx – k)(x – (fx + k)/2)

Variables Table

Variable	Meaning	Unit	Typical Range
(fx, fy)	Coordinates of the focus	Units	Any real numbers
y = k or x = k	Equation of the directrix	Units	Any real number for k
(h, v)	Coordinates of the vertex	Units	Calculated
p	Distance from vertex to focus/directrix	Units	Non-zero real number
4p	Latus rectum length	Units	Calculated

The find the parabola with focus and directrix calculator uses these formulas to derive the equation.

Practical Examples (Real-World Use Cases)

Example 1: Satellite Dish Design

An engineer is designing a satellite dish. The focus is located at (0, 2) and the directrix is the line y = -2. Using the find the parabola with focus and directrix calculator:

• Focus: (0, 2)
• Directrix: y = -2
• The calculator finds: Vertex (0, 0), p = 2, Equation: x² = 8y. This describes the shape of the dish reflector.

Example 2: Headlight Reflector

A car headlight reflector has its light source (focus) at (1.5, 0) and the directrix is x = -1.5. Using the find the parabola with focus and directrix calculator:

• Focus: (1.5, 0)
• Directrix: x = -1.5
• The calculator finds: Vertex (0, 0), p = 1.5, Equation: y² = 6x. This parabolic shape reflects light from the focus into a parallel beam.

You can also use our parabola vertex calculator for related calculations.

How to Use This Find the Parabola with Focus and Directrix Calculator

1. Enter Focus Coordinates: Input the x and y coordinates of the focus point (fx, fy).
2. Select Directrix Type: Choose whether the directrix is a horizontal line (y = k) or a vertical line (x = k) from the dropdown.
3. Enter Directrix Value: Input the value of ‘k’ from the directrix equation.
4. Calculate: The calculator automatically updates as you enter values, or you can click “Calculate”.
5. Review Results: The calculator displays the equation of the parabola in standard form, the coordinates of the vertex, the value of ‘p’, and the axis of symmetry. It also provides a visual representation and a summary table.

Understanding the results helps in visualizing the parabola’s orientation and position. The value and sign of ‘p’ tell you the direction the parabola opens and the distance from the vertex to the focus. Check out our graphing calculator to visualize the equation.

Key Factors That Affect Find the Parabola with Focus and Directrix Calculator Results

• Focus Coordinates (fx, fy): Changing the focus position shifts the entire parabola and its vertex. The focus directly influences the ‘h’ and ‘v’ of the vertex and ‘p’.
• Directrix Equation (y=k or x=k): The type of directrix (horizontal or vertical) determines whether the parabola opens up/down or left/right. The value of ‘k’ affects the position of the vertex and the value of ‘p’.
• Relative Position of Focus and Directrix: The distance between the focus and the directrix determines the magnitude of ‘p’ (which is half this distance) and thus how “wide” or “narrow” the parabola is. If the focus is above the y=k directrix, p>0, opens up.
• The value of ‘p’: Calculated as half the distance between focus and directrix, ‘p’ dictates the width of the parabola at the focus (latus rectum = |4p|). A larger |p| means a wider parabola.
• Axis of Symmetry: This line passes through the focus and vertex, perpendicular to the directrix. Its position is determined by the fixed coordinate of the focus and vertex (x=fx for y=k, y=fy for x=k).
• Vertex Position: The vertex is the midpoint between the focus and the point on the directrix closest to the focus. Its position is entirely dependent on the focus and directrix. See our midpoint calculator for more on midpoints.

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: How does the find the parabola with focus and directrix calculator work?

A: It uses the definition of a parabola and the distance formula to derive the standard equation based on the given focus and directrix coordinates/equation.

Q: What does ‘p’ represent?

A: ‘p’ is the directed distance from the vertex to the focus (and from the vertex to the directrix). Its absolute value is the distance, and its sign indicates the direction the parabola opens relative to the vertex.

Q: Can the focus be on the directrix?

A: No, if the focus were on the directrix, the parabola would degenerate into a line. p would be 0.

Q: What if the directrix is y=k?

A: The parabola opens up or down, and its axis of symmetry is vertical (x=fx).

Q: What if the directrix is x=k?

A: The parabola opens left or right, and its axis of symmetry is horizontal (y=fy).

Q: How is the vertex related to the focus and directrix?

A: The vertex is the midpoint between the focus and the directrix, lying on the axis of symmetry.

Q: Can I use this calculator for any parabola?

A: Yes, as long as you know the focus and directrix, this find the parabola with focus and directrix calculator can find the equation of any parabola not rotated.

Related Tools and Internal Resources