Find Foci Of Parabola Calculator

Easily calculate the focus (foci) of a parabola using its vertex form equation with our Find Foci of Parabola Calculator. Input the parameters and get instant results.

What is a Parabola's Focus (Foci)?

The focus (plural: foci) of a parabola is a special point located on the axis of symmetry of the parabola. Its position is crucial in defining the parabola's shape and properties. A parabola is geometrically defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed line (the directrix). This Find Foci of Parabola Calculator helps you locate this point precisely.

The distance from the vertex of the parabola to the focus is denoted by 'a' (the focal length). The focus lies inside the curve of the parabola. This property is used in many applications, such as satellite dishes, car headlights, and telescopes, where the focus is used to collect or direct waves (light, sound, radio).

Anyone studying conic sections, physics (optics, antennas), or engineering might need to use a find foci of parabola calculator to determine the focus of a parabola given its equation.

A common misconception is that 'foci' always implies two points; however, a parabola has only one focus. Ellipses and hyperbolas have two foci.

Find Foci of Parabola Calculator: Formula and Mathematical Explanation

The standard vertex form equations for a parabola are:

• (x – h)² = 4a(y – k): This parabola opens vertically (up or down).
• (y – k)² = 4a(x – h): This parabola opens horizontally (left or right).

In these equations:

• (h, k) are the coordinates of the vertex.
• 'a' is the distance from the vertex to the focus and from the vertex to the directrix. The sign of '4a' determines the direction of opening relative to the standard orientation.

For (x – h)² = 4a(y – k):

• The parabola opens upwards if 4a > 0 (a > 0).
• The parabola opens downwards if 4a < 0 (a < 0).
• The axis of symmetry is x = h.
• The focus is located at (h, k + a).
• The directrix is the line y = k – a.

For (y – k)² = 4a(x – h):

• The parabola opens to the right if 4a > 0 (a > 0).
• The parabola opens to the left if 4a < 0 (a < 0).
• The axis of symmetry is y = k.
• The focus is located at (h + a, k).
• The directrix is the line x = h – a.

Our find foci of parabola calculator uses these formulas to determine the focus based on your inputs for h, k, and 4a, and the selected equation form.

Variables Table

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
4a	Coefficient multiplying the linear term (latus rectum length)	(length units)	Any non-zero real number
a	Focal length (distance from vertex to focus)	(length units)	Any non-zero real number

Practical Examples (Real-World Use Cases)

Example 1: Parabolic Reflector

A satellite dish is shaped like a paraboloid, which is a 3D parabola. Its cross-section is a parabola. Suppose the equation of the cross-section is (x – 0)² = 12(y – 0), with the vertex at (0, 0). We want to find the focus where the receiver should be placed.

• Equation form: (x – h)² = 4a(y – k)
• h = 0, k = 0, 4a = 12
• a = 12 / 4 = 3
• Since 4a > 0, it opens upwards.
• Focus = (h, k + a) = (0, 0 + 3) = (0, 3)

The receiver should be placed 3 units above the vertex along the axis of symmetry. You can verify this with the find foci of parabola calculator.

Example 2: Headlight Reflector

The reflector in a car headlight has a parabolic cross-section. If the vertex is at (-2, 0) and the parabola's equation is (y – 0)² = -8(x – (-2)), or y² = -8(x + 2), where is the focus (where the bulb should be)?

• Equation form: (y – k)² = 4a(x – h)
• h = -2, k = 0, 4a = -8
• a = -8 / 4 = -2
• Since 4a < 0, it opens to the left.
• Focus = (h + a, k) = (-2 + (-2), 0) = (-4, 0)

The bulb (focus) should be at (-4, 0). The find foci of parabola calculator can quickly give you this.

How to Use This Find Foci of Parabola Calculator

1. Select Equation Form: Choose whether your parabola's equation is in the form (x – h)² = 4a(y – k) or (y – k)² = 4a(x – h).
2. Enter Vertex Coordinates (h, k): Input the values for 'h' and 'k' from your equation.
3. Enter the Value of 4a: Input the coefficient '4a' from your equation. Ensure it's not zero.
4. Calculate: The calculator automatically updates, or you can click "Calculate".
5. Read Results:
   • Primary Result: The coordinates of the focus (foci for other conics, but focus for parabola).
   • Intermediate Results: The vertex, the value of 'a', the direction the parabola opens, and the equation of the directrix.
   • Graph: A visual representation of the parabola, vertex, focus, and directrix.
6. Reset: Click "Reset" to return to default values.
7. Copy: Click "Copy Results" to copy the focus, vertex, 'a', direction, and directrix to your clipboard.

Our find foci of parabola calculator is designed for ease of use and accuracy.

Key Factors That Affect Parabola Focus Results

The location of the focus of a parabola is determined by:

1. Vertex Coordinates (h, k): The focus is located relative to the vertex. If the vertex shifts, the focus shifts with it.
2. Value and Sign of 'a' (or 4a): The magnitude of 'a' determines the distance between the vertex and the focus (focal length). A larger |a| means the focus is further from the vertex, and the parabola is wider. The sign of 'a' (or 4a), combined with the equation form, determines the direction the parabola opens and thus the direction of the focus from the vertex.
3. Equation Form (which variable is squared): This determines whether the parabola opens vertically or horizontally, changing whether 'a' is added to 'k' or 'h' to find the focus.
4. Axis of Symmetry: The focus always lies on the axis of symmetry. For (x-h)²=4a(y-k), it's x=h; for (y-k)²=4a(x-h), it's y=k.
5. Directrix: The focus and directrix are equidistant from the vertex, on opposite sides. The directrix is perpendicular to the axis of symmetry.
6. Latus Rectum: The length of the latus rectum is |4a|, which is the width of the parabola at the focus, perpendicular to the axis of symmetry.

Understanding these factors helps in interpreting the results from the find foci of parabola calculator and the parabola's geometry.

Frequently Asked Questions (FAQ)

What is the focus of a parabola?

The focus is a fixed point inside the parabola from which all points on the parabola are equidistant to it and a line called the directrix.

How many foci does a parabola have?

A parabola has only one focus. Ellipses and hyperbolas have two foci.

What is 'a' in the parabola equation?

'a' represents the distance from the vertex to the focus and from the vertex to the directrix. Its sign indicates direction relative to the vertex along the axis of symmetry.

How does the find foci of parabola calculator work?

It uses the vertex form of the parabola equation (either (x-h)²=4a(y-k) or (y-k)²=4a(x-h)) and the input values of h, k, and 4a to calculate the coordinates of the focus using the formulas Focus(h, k+a) or Focus(h+a, k) respectively, where a = 4a/4.

What if 4a is zero?

If 4a is zero, the equation does not represent a parabola but degenerates into lines or no graph. The calculator requires a non-zero 4a.

Can the focus be the same as the vertex?

No, the focus is always a distance |a| away from the vertex, and 'a' cannot be zero for a parabola.

What does the sign of 4a tell me?

If the equation is (x-h)²=4a(y-k), 4a > 0 means opens up, 4a < 0 means opens down. If (y-k)²=4a(x-h), 4a > 0 means opens right, 4a < 0 means opens left.

Where is the directrix located?

The directrix is a line y = k – a for (x-h)²=4a(y-k) and x = h – a for (y-k)²=4a(x-h). Our find foci of parabola calculator also provides the directrix equation.

Related Tools and Internal Resources

• Parabola Equation Calculator: Find the equation of a parabola given certain properties.
• Vertex Calculator: Calculate the vertex of a parabola from its equation.
• Directrix Calculator: Find the equation of the directrix of a parabola.
• Graphing Parabolas Tool: Interactively graph parabolas.
• Conic Sections Calculator: Explore properties of circles, ellipses, parabolas, and hyperbolas.
• Distance Formula Calculator: Calculate the distance between two points.

These tools, including the find foci of parabola calculator, can help you further explore the properties of parabolas and other conic sections.